Theorem th3qlem2 3251

Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption.

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⟩)))

Assertion

Ref	Expression
th3qlem2	⊢ ((A ∈ ((S × S) / R) ∧ B ∈ ((S × S) / R)) → ∃*z∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))

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

Proof of Theorem th3qlem2

Step	Hyp	Ref	Expression
1		th3q.2	. . 3 ⊢ Er R
2		th3q.3	. . 3 ⊢ dom R = (S × S)
3		cleqid 1102	. . . . 5 ⊢ (S × S) = (S × S)
4		breq1 2065	. . . . . . . 8 ⊢ (⟨w, v⟩ = s → (⟨w, v⟩R⟨u, t⟩ ↔ sR⟨u, t⟩))
5	4	anbi1d 469	. . . . . . 7 ⊢ (⟨w, v⟩ = s → ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) ↔ (sR⟨u, t⟩ ∧ xRy)))
6		opreq1 3006	. . . . . . . 8 ⊢ (⟨w, v⟩ = s → (⟨w, v⟩Fx) = (sFx))
7	6	breq1d 2071	. . . . . . 7 ⊢ (⟨w, v⟩ = s → ((⟨w, v⟩Fx)R(⟨u, t⟩Fy) ↔ (sFx)R(⟨u, t⟩Fy)))
8	5, 7	imbi12d 474	. . . . . 6 ⊢ (⟨w, v⟩ = s → (((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy)) ↔ ((sR⟨u, t⟩ ∧ xRy) → (sFx)R(⟨u, t⟩Fy))))
9	8	imbi2d 464	. . . . 5 ⊢ (⟨w, v⟩ = s → (((x ∈ (S × S) ∧ y ∈ (S × S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy))) ↔ ((x ∈ (S × S) ∧ y ∈ (S × S)) → ((sR⟨u, t⟩ ∧ xRy) → (sFx)R(⟨u, t⟩Fy)))))
10		breq2 2066	. . . . . . . 8 ⊢ (⟨u, t⟩ = f → (sR⟨u, t⟩ ↔ sRf))
11	10	anbi1d 469	. . . . . . 7 ⊢ (⟨u, t⟩ = f → ((sR⟨u, t⟩ ∧ xRy) ↔ (sRf ∧ xRy)))
12		opreq1 3006	. . . . . . . 8 ⊢ (⟨u, t⟩ = f → (⟨u, t⟩Fy) = (fFy))
13	12	breq2d 2072	. . . . . . 7 ⊢ (⟨u, t⟩ = f → ((sFx)R(⟨u, t⟩Fy) ↔ (sFx)R(fFy)))
14	11, 13	imbi12d 474	. . . . . 6 ⊢ (⟨u, t⟩ = f → (((sR⟨u, t⟩ ∧ xRy) → (sFx)R(⟨u, t⟩Fy)) ↔ ((sRf ∧ xRy) → (sFx)R(fFy))))
15	14	imbi2d 464	. . . . 5 ⊢ (⟨u, t⟩ = f → (((x ∈ (S × S) ∧ y ∈ (S × S)) → ((sR⟨u, t⟩ ∧ xRy) → (sFx)R(⟨u, t⟩Fy))) ↔ ((x ∈ (S × S) ∧ y ∈ (S × S)) → ((sRf ∧ xRy) → (sFx)R(fFy)))))
16		breq1 2065	. . . . . . . . . 10 ⊢ (⟨s, f⟩ = x → (⟨s, f⟩R⟨g, h⟩ ↔ xR⟨g, h⟩))
17	16	anbi2d 468	. . . . . . . . 9 ⊢ (⟨s, f⟩ = x → ((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) ↔ (⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩)))
18		opreq2 3007	. . . . . . . . . 10 ⊢ (⟨s, f⟩ = x → (⟨w, v⟩F⟨s, f⟩) = (⟨w, v⟩Fx))
19	18	breq1d 2071	. . . . . . . . 9 ⊢ (⟨s, f⟩ = x → ((⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩) ↔ (⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩)))
20	17, 19	imbi12d 474	. . . . . . . 8 ⊢ (⟨s, f⟩ = x → (((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) → (⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩)) ↔ ((⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩) → (⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩))))
21	20	imbi2d 464	. . . . . . 7 ⊢ (⟨s, f⟩ = x → ((((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) → (⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩))) ↔ (((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩) → (⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩)))))
22		breq2 2066	. . . . . . . . . 10 ⊢ (⟨g, h⟩ = y → (xR⟨g, h⟩ ↔ xRy))
23	22	anbi2d 468	. . . . . . . . 9 ⊢ (⟨g, h⟩ = y → ((⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩) ↔ (⟨w, v⟩R⟨u, t⟩ ∧ xRy)))
24		opreq2 3007	. . . . . . . . . 10 ⊢ (⟨g, h⟩ = y → (⟨u, t⟩F⟨g, h⟩) = (⟨u, t⟩Fy))
25	24	breq2d 2072	. . . . . . . . 9 ⊢ (⟨g, h⟩ = y → ((⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩) ↔ (⟨w, v⟩Fx)R(⟨u, t⟩Fy)))
26	23, 25	imbi12d 474	. . . . . . . 8 ⊢ (⟨g, h⟩ = y → (((⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩) → (⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩)) ↔ ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy))))
27	26	imbi2d 464	. . . . . . 7 ⊢ (⟨g, h⟩ = y → ((((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xR⟨g, h⟩) → (⟨w, v⟩Fx)R(⟨u, t⟩F⟨g, h⟩))) ↔ (((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy)))))
28		th3q.4	. . . . . . . . 9 ⊢ ((((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⟩)))
29	28	exp 291	. . . . . . . 8 ⊢ (((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⟩))))
30	29	com12 13	. . . . . . 7 ⊢ (((s ∈ S ∧ f ∈ S) ∧ (g ∈ S ∧ h ∈ S)) → (((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ ⟨s, f⟩R⟨g, h⟩) → (⟨w, v⟩F⟨s, f⟩)R(⟨u, t⟩F⟨g, h⟩))))
31	3, 21, 27, 30	2optocl 2470	. . . . . 6 ⊢ ((x ∈ (S × S) ∧ y ∈ (S × S)) → (((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy))))
32	31	com12 13	. . . . 5 ⊢ (((w ∈ S ∧ v ∈ S) ∧ (u ∈ S ∧ t ∈ S)) → ((x ∈ (S × S) ∧ y ∈ (S × S)) → ((⟨w, v⟩R⟨u, t⟩ ∧ xRy) → (⟨w, v⟩Fx)R(⟨u, t⟩Fy))))
33	3, 9, 15, 32	2optocl 2470	. . . 4 ⊢ ((s ∈ (S × S) ∧ f ∈ (S × S)) → ((x ∈ (S × S) ∧ y ∈ (S × S)) → ((sRf ∧ xRy) → (sFx)R(fFy))))
34	33	imp 277	. . 3 ⊢ (((s ∈ (S × S) ∧ f ∈ (S × S)) ∧ (x ∈ (S × S) ∧ y ∈ (S × S))) → ((sRf ∧ xRy) → (sFx)R(fFy)))
35	1, 2, 34	th3qlem1 3250	. 2 ⊢ ((A ∈ ((S × S) / R) ∧ B ∈ ((S × S) / R)) → ∃*z∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R))
36		immo 1043	. . 3 ⊢ (∀z(∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) → ∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R)) → (∃*z∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R) → ∃*z∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
37		opex 1893	. . . . . 6 ⊢ ⟨w, v⟩ ∈ V
38		opex 1893	. . . . . 6 ⊢ ⟨u, t⟩ ∈ V
39		eceq2 3215	. . . . . . . . 9 ⊢ (s = ⟨w, v⟩ → [s]R = [⟨w, v⟩]R)
40	39	cleq2d 1112	. . . . . . . 8 ⊢ (s = ⟨w, v⟩ → (A = [s]R ↔ A = [⟨w, v⟩]R))
41		eceq2 3215	. . . . . . . . 9 ⊢ (x = ⟨u, t⟩ → [x]R = [⟨u, t⟩]R)
42	41	cleq2d 1112	. . . . . . . 8 ⊢ (x = ⟨u, t⟩ → (B = [x]R ↔ B = [⟨u, t⟩]R))
43	40, 42	bi2anan9 478	. . . . . . 7 ⊢ ((s = ⟨w, v⟩ ∧ x = ⟨u, t⟩) → ((A = [s]R ∧ B = [x]R) ↔ (A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R)))
44		opreq12 3008	. . . . . . . . 9 ⊢ ((s = ⟨w, v⟩ ∧ x = ⟨u, t⟩) → (sFx) = (⟨w, v⟩F⟨u, t⟩))
45		eceq2 3215	. . . . . . . . 9 ⊢ ((sFx) = (⟨w, v⟩F⟨u, t⟩) → [(sFx)]R = [(⟨w, v⟩F⟨u, t⟩)]R)
46	44, 45	syl 12	. . . . . . . 8 ⊢ ((s = ⟨w, v⟩ ∧ x = ⟨u, t⟩) → [(sFx)]R = [(⟨w, v⟩F⟨u, t⟩)]R)
47	46	cleq2d 1112	. . . . . . 7 ⊢ ((s = ⟨w, v⟩ ∧ x = ⟨u, t⟩) → (z = [(sFx)]R ↔ z = [(⟨w, v⟩F⟨u, t⟩)]R))
48	43, 47	anbi12d 476	. . . . . 6 ⊢ ((s = ⟨w, v⟩ ∧ x = ⟨u, t⟩) → (((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R) ↔ ((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R)))
49	37, 38, 48	cla4e2v 1406	. . . . 5 ⊢ (((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) → ∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R))
50	49	19.23aivv 953	. . . 4 ⊢ (∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) → ∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R))
51	50	19.23aivv 953	. . 3 ⊢ (∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R) → ∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R))
52	36, 51	mpg 684	. 2 ⊢ (∃*z∃s∃x((A = [s]R ∧ B = [x]R) ∧ z = [(sFx)]R) → ∃*z∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))
53	35, 52	syl 12	1 ⊢ ((A ∈ ((S × S) / R) ∧ B ∈ ((S × S) / R)) → ∃*z∃w∃v∃u∃t((A = [⟨w, v⟩]R ∧ B = [⟨u, t⟩]R) ∧ z = [(⟨w, v⟩F⟨u, t⟩)]R))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054   × cxp 2408  dom cdm 2410  (class class class)co 3001  Er wer 3197  [cec 3198   / cqs 3199

This theorem is referenced by:  th3qcor 3252  th3q 3253

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-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-er 3200  df-ec 3202  df-qs 3205

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