Theorem oprabval3 3052

Description: The value of an operation abstraction. Special case.

Hypotheses

Ref	Expression
oprabval3.1	⊢ S ∈ V
oprabval3.2	⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → R = S)
oprabval3.3	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}

Assertion

Ref	Expression
oprabval3	⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (⟨A, B⟩F⟨C, D⟩) = S)

Distinct variable group(s):   x,y,z,w,v,u,f,A   x,B,y,z,w,v,u,f   x,C,y,z,w,v,u,f   x,D,y,z,w,v,u,f   x,H,y,z,w,v,u,f   x,R,y,z   x,S,y,z,w,v,u,f

Proof of Theorem oprabval3

Step	Hyp	Ref	Expression
1		oprabval3.1	. . 3 ⊢ S ∈ V
2		cleq1 1107	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (x = ⟨w, v⟩ ↔ ⟨A, B⟩ = ⟨w, v⟩))
3	2	anbi1d 469	. . . . 5 ⊢ (x = ⟨A, B⟩ → ((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩)))
4	3	anbi1d 469	. . . 4 ⊢ (x = ⟨A, B⟩ → (((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)))
5	4	bi4exdv 940	. . 3 ⊢ (x = ⟨A, B⟩ → (∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)))
6		cleq1 1107	. . . . . 6 ⊢ (y = ⟨C, D⟩ → (y = ⟨u, f⟩ ↔ ⟨C, D⟩ = ⟨u, f⟩))
7	6	anbi2d 468	. . . . 5 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩)))
8	7	anbi1d 469	. . . 4 ⊢ (y = ⟨C, D⟩ → (((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R)))
9	8	bi4exdv 940	. . 3 ⊢ (y = ⟨C, D⟩ → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R)))
10		cleq1 1107	. . . . 5 ⊢ (z = S → (z = R ↔ S = R))
11	10	anbi2d 468	. . . 4 ⊢ (z = S → (((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) ↔ ((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ S = R)))
12	11	bi4exdv 940	. . 3 ⊢ (z = S → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ S = R)))
13		moeq 1431	. . . . . . 7 ⊢ ∃*z z = R
14	13	mosubop 1911	. . . . . 6 ⊢ ∃*z∃u∃f(y = ⟨u, f⟩ ∧ z = R)
15	14	mosubop 1911	. . . . 5 ⊢ ∃*z∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R))
16		anass 336	. . . . . . . . 9 ⊢ (((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
17	16	bi2ex 734	. . . . . . . 8 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
18		19.42vv 968	. . . . . . . 8 ⊢ (∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
19	17, 18	bitr 151	. . . . . . 7 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
20	19	bi2ex 734	. . . . . 6 ⊢ (∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
21	20	bimo 1031	. . . . 5 ⊢ (∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃*z∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
22	15, 21	mpbir 165	. . . 4 ⊢ ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)
23	22	a1i 7	. . 3 ⊢ ((x ∈ (H × H) ∧ y ∈ (H × H)) → ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))
24		oprabval3.3	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
25	1, 5, 9, 12, 23, 24	oprabvali 3049	. 2 ⊢ ((⟨A, B⟩ ∈ (H × H) ∧ ⟨C, D⟩ ∈ (H × H)) → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ S = R) → (⟨A, B⟩F⟨C, D⟩) = S))
26		opelxpi 2455	. . 3 ⊢ ((A ∈ H ∧ B ∈ H) → ⟨A, B⟩ ∈ (H × H))
27		opelxpi 2455	. . 3 ⊢ ((C ∈ H ∧ D ∈ H) → ⟨C, D⟩ ∈ (H × H))
28	26, 27	anim12i 268	. 2 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (⟨A, B⟩ ∈ (H × H) ∧ ⟨C, D⟩ ∈ (H × H)))
29		cleqid 1102	. . 3 ⊢ S = S
30		oprabval3.2	. . . . 5 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → R = S)
31	30	cleq2d 1112	. . . 4 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → (S = R ↔ S = S))
32	31	copsex4g 1904	. . 3 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ S = R) ↔ S = S))
33	29, 32	mpbiri 169	. 2 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → ∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ S = R))
34	25, 28, 33	sylc 62	1 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (⟨A, B⟩F⟨C, D⟩) = S)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   × cxp 2408  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  oprec 3254  addcnsr 4047  mulcnsr 4048

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-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

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