Theorem th3q 3253

Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs.

Hypotheses

Ref	Expression
th3q.1	⊢ R ∈ V
th3q.2	⊢ Er R
th3q.3	⊢ dom R = (S × S)
th3q.4	⊢ ((((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) ∧ ((s ∈ S ∧ f ∈ S) ∧ (g ∈ S ∧ h ∈ S))) → ((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) → (⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩)))
th3q.5	⊢ G = {⟨⟨x, y⟩, z⟩∣((x ∈ ((S × S) / R) ∧ y ∈ ((S × S) / R)) ∧ ∃w∃v∃u∃t((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))}

Assertion

Ref	Expression
th3q	⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → ([⟨A, B⟩]RG[⟨C, D⟩]R) = [(⟨A, B⟩F⟨C, D⟩)]R)

Distinct variable group(s):   x,y,z,w,v,u,t,s,f,g,h,R   x,S,y,z,w,v,u,t,s,f,g,h   x,A,y,z,w,v,u,t,s,f,g,h   x,B,y,z,w,v,u,t,s,f,g,h   x,C,y,z,w,v,u,t,s,f,g,h   x,D,y,z,w,v,u,t,s,f,g,h   x,F,y,z,w,v,u,t,s,f,g,h   x,G,y,z,w,v,u,t,s,f,g,h

Proof of Theorem th3q

Step	Hyp	Ref	Expression
1		th3q.1	. . . 4 ⊢ R ∈ V
2		ecexg 3204	. . . 4 ⊢ (R ∈ V → [(⟨A, B⟩F⟨C, D⟩)]R ∈ V)
3	1, 2	ax-mp 6	. . 3 ⊢ [(⟨A, B⟩F⟨C, D⟩)]R ∈ V
4		cleq1 1107	. . . . . 6 ⊢ (x = [⟨A, B⟩]R → (x = [⟨w, v⟩]R ↔ [⟨A, B⟩]R = [⟨w, v⟩]R))
5	4	anbi1d 469	. . . . 5 ⊢ (x = [⟨A, B⟩]R → ((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ↔ ([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R)))
6	5	anbi1d 469	. . . 4 ⊢ (x = [⟨A, B⟩]R → (((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
7	6	bi4exdv 940	. . 3 ⊢ (x = [⟨A, B⟩]R → (∃w∃v∃u∃t((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
8		cleq1 1107	. . . . . 6 ⊢ (y = [⟨C, D⟩]R → (y = [⟨u, t⟩]R ↔ [⟨C, D⟩]R = [⟨u, t⟩]R))
9	8	anbi2d 468	. . . . 5 ⊢ (y = [⟨C, D⟩]R → (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ↔ ([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R)))
10	9	anbi1d 469	. . . 4 ⊢ (y = [⟨C, D⟩]R → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
11	10	bi4exdv 940	. . 3 ⊢ (y = [⟨C, D⟩]R → (∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
12		cleq1 1107	. . . . 5 ⊢ (z = [(⟨A, B⟩F⟨C, D⟩)]R → (z = [(⟨w, v⟩F⟨u, t⟩)]R ↔ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R))
13	12	anbi2d 468	. . . 4 ⊢ (z = [(⟨A, B⟩F⟨C, D⟩)]R → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R)))
14	13	bi4exdv 940	. . 3 ⊢ (z = [(⟨A, B⟩F⟨C, D⟩)]R → (∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R)))
15		th3q.2	. . . 4 ⊢ Er R
16		th3q.3	. . . 4 ⊢ dom R = (S × S)
17		th3q.4	. . . 4 ⊢ ((((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) ∧ ((s ∈ S ∧ f ∈ S) ∧ (g ∈ S ∧ h ∈ S))) → ((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) → (⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩)))
18	1, 15, 16, 17	th3qlem2 3251	. . 3 ⊢ ((x ∈ ((S × S) / R) ∧ y ∈ ((S × S) / R)) → ∃*z∃w∃v∃u∃t((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))
19		th3q.5	. . 3 ⊢ G = {⟨⟨x, y⟩, z⟩∣((x ∈ ((S × S) / R) ∧ y ∈ ((S × S) / R)) ∧ ∃w∃v∃u∃t((x = [⟨w, v⟩]R ∧ y = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))}
20	3, 7, 11, 14, 18, 19	oprabvali 3049	. 2 ⊢ (([⟨A, B⟩]R ∈ ((S × S) / R) ∧ [⟨C, D⟩]R ∈ ((S × S) / R)) → (∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R) → ([⟨A, B⟩]RG[⟨C, D⟩]R) = [(⟨A, B⟩F⟨C, D⟩)]R))
21		opelxpi 2455	. . . 4 ⊢ ((A ∈ S ∧ B ∈ S) → ⟨A, B⟩ ∈ (S × S))
22	1	ecelqsi 3229	. . . 4 ⊢ (⟨A, B⟩ ∈ (S × S) → [⟨A, B⟩]R ∈ ((S × S) / R))
23	21, 22	syl 12	. . 3 ⊢ ((A ∈ S ∧ B ∈ S) → [⟨A, B⟩]R ∈ ((S × S) / R))
24		opelxpi 2455	. . . 4 ⊢ ((C ∈ S ∧ D ∈ S) → ⟨C, D⟩ ∈ (S × S))
25	1	ecelqsi 3229	. . . 4 ⊢ (⟨C, D⟩ ∈ (S × S) → [⟨C, D⟩]R ∈ ((S × S) / R))
26	24, 25	syl 12	. . 3 ⊢ ((C ∈ S ∧ D ∈ S) → [⟨C, D⟩]R ∈ ((S × S) / R))
27	23, 26	anim12i 268	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → ([⟨A, B⟩]R ∈ ((S × S) / R) ∧ [⟨C, D⟩]R ∈ ((S × S) / R)))
28		cleqid 1102	. . . 4 ⊢ [⟨A, B⟩]R = [⟨A, B⟩]R
29		cleqid 1102	. . . 4 ⊢ [⟨C, D⟩]R = [⟨C, D⟩]R
30	28, 29	pm3.2i 234	. . 3 ⊢ ([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R)
31		cleqid 1102	. . 3 ⊢ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R
32		opeq12 1878	. . . . . 6 ⊢ ((w = A ∧ v = B) → ⟨w, v⟩ = ⟨A, B⟩)
33		eceq2 3215	. . . . . . . . 9 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → [⟨w, v⟩]R = [⟨A, B⟩]R)
34	33	cleq2d 1112	. . . . . . . 8 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → ([⟨A, B⟩]R = [⟨w, v⟩]R ↔ [⟨A, B⟩]R = [⟨A, B⟩]R))
35	34	anbi1d 469	. . . . . . 7 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ↔ ([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R)))
36		opreq1 3006	. . . . . . . . 9 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → (⟨w, v⟩F⟨C, D⟩) = (⟨A, B⟩F⟨C, D⟩))
37		eceq2 3215	. . . . . . . . 9 ⊢ ((⟨w, v⟩F⟨C, D⟩) = (⟨A, B⟩F⟨C, D⟩) → [(⟨w, v⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R)
38	36, 37	syl 12	. . . . . . . 8 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → [(⟨w, v⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R)
39	38	cleq2d 1112	. . . . . . 7 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → ([(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R ↔ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R))
40	35, 39	anbi12d 476	. . . . . 6 ⊢ (⟨w, v⟩ = ⟨A, B⟩ → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R) ↔ (([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R)))
41	32, 40	syl 12	. . . . 5 ⊢ ((w = A ∧ v = B) → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R) ↔ (([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R)))
42	41	cla4e2gv 1398	. . . 4 ⊢ ((A ∈ S ∧ B ∈ S) → ((([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R) → ∃w∃v(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R)))
43		opeq12 1878	. . . . . . 7 ⊢ ((u = C ∧ t = D) → ⟨u, t⟩ = ⟨C, D⟩)
44		eceq2 3215	. . . . . . . . . 10 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → [⟨u, t⟩]R = [⟨C, D⟩]R)
45	44	cleq2d 1112	. . . . . . . . 9 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → ([⟨C, D⟩]R = [⟨u, t⟩]R ↔ [⟨C, D⟩]R = [⟨C, D⟩]R))
46	45	anbi2d 468	. . . . . . . 8 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ↔ ([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R)))
47		opreq2 3007	. . . . . . . . . 10 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → (⟨w, v⟩F⟨u, t⟩) = (⟨w, v⟩F⟨C, D⟩))
48		eceq2 3215	. . . . . . . . . 10 ⊢ ((⟨w, v⟩F⟨u, t⟩) = (⟨w, v⟩F⟨C, D⟩) → [(⟨w, v⟩F⟨u, t⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R)
49	47, 48	syl 12	. . . . . . . . 9 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → [(⟨w, v⟩F⟨u, t⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R)
50	49	cleq2d 1112	. . . . . . . 8 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → ([(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R ↔ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R))
51	46, 50	anbi12d 476	. . . . . . 7 ⊢ (⟨u, t⟩ = ⟨C, D⟩ → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R)))
52	43, 51	syl 12	. . . . . 6 ⊢ ((u = C ∧ t = D) → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R) ↔ (([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R)))
53	52	cla4e2gv 1398	. . . . 5 ⊢ ((C ∈ S ∧ D ∈ S) → ((([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R) → ∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R)))
54	53	19.22dvv 949	. . . 4 ⊢ ((C ∈ S ∧ D ∈ S) → (∃w∃v(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨C, D⟩)]R) → ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R)))
55	42, 54	sylan9 359	. . 3 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → ((([⟨A, B⟩]R = [⟨A, B⟩]R ∧ [⟨C, D⟩]R = [⟨C, D⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨A, B⟩F⟨C, D⟩)]R) → ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R)))
56	30, 31, 55	mp2ani 523	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → ∃w∃v∃u∃t(([⟨A, B⟩]R = [⟨w, v⟩]R ∧ [⟨C, D⟩]R = [⟨u, t⟩]R) ∧ [(⟨A, B⟩F⟨C, D⟩)]R = [(⟨w, v⟩F⟨u, t⟩)]R))
57	20, 27, 56	sylc 62	1 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → ([⟨A, B⟩]RG[⟨C, D⟩]R) = [(⟨A, B⟩F⟨C, D⟩)]R)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054   × cxp 2408  dom cdm 2410  (class class class)co 3001  {copab2 3002  Er wer 3197  [cec 3198   / cqs 3199

This theorem is referenced by:  oprec 3254

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-er 3200  df-ec 3202  df-qs 3205

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