Theorem ov3 5600

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ov3.1	⊢ S ∈ V
ov3.2	⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → R = S)
ov3.3	⊢ 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
ov3	⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (⟨A, B⟩F⟨C, D⟩) = S)

Distinct variable groups:   u,f,v,w,x,y,z,A   B,f,u,v,w,x,y,z   x,R,y,z   C,f,u,v,w,y,z   D,f,u,v,w,y,z   f,H,u,v,w,x,y,z   S,f,u,v,w,z

Allowed substitution hints:   C(x)   D(x)   R(w, v, u, f)   S(x, y)   F(x, y, z, w, v, u, f)

Proof of Theorem ov3

Step	Hyp	Ref	Expression
1		ov3.1	. . 3 ⊢ S ∈ V
2	1	isseti 2866	. 2 ⊢ ∃z z = S
3		nfv 1619	. . 3 ⊢ Ⅎz((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H))
4		nfcv 2490	. . . . 5 ⊢ Ⅎz⟨A, B⟩
5		ov3.3	. . . . . 6 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
6		nfoprab3 5549	. . . . . 6 ⊢ Ⅎz{⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
7	5, 6	nfcxfr 2487	. . . . 5 ⊢ ℲzF
8		nfcv 2490	. . . . 5 ⊢ Ⅎz⟨C, D⟩
9	4, 7, 8	nfov 5546	. . . 4 ⊢ Ⅎz(⟨A, B⟩F⟨C, D⟩)
10	9	nfeq1 2499	. . 3 ⊢ Ⅎz(⟨A, B⟩F⟨C, D⟩) = S
11		ov3.2	. . . . . . 7 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → R = S)
12	11	eqeq2d 2364	. . . . . 6 ⊢ (((w = A ∧ v = B) ∧ (u = C ∧ f = D)) → (z = R ↔ z = S))
13	12	copsex4g 4611	. . . . 5 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) ↔ z = S))
14		opelxp 4812	. . . . . 6 ⊢ (⟨A, B⟩ ∈ (H × H) ↔ (A ∈ H ∧ B ∈ H))
15		opelxp 4812	. . . . . 6 ⊢ (⟨C, D⟩ ∈ (H × H) ↔ (C ∈ H ∧ D ∈ H))
16		nfcv 2490	. . . . . . 7 ⊢ Ⅎx⟨A, B⟩
17		nfcv 2490	. . . . . . 7 ⊢ Ⅎy⟨A, B⟩
18		nfcv 2490	. . . . . . 7 ⊢ Ⅎy⟨C, D⟩
19		nfv 1619	. . . . . . . 8 ⊢ Ⅎx∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)
20		nfoprab1 5547	. . . . . . . . . . 11 ⊢ Ⅎx{⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
21	5, 20	nfcxfr 2487	. . . . . . . . . 10 ⊢ ℲxF
22		nfcv 2490	. . . . . . . . . 10 ⊢ Ⅎxy
23	16, 21, 22	nfov 5546	. . . . . . . . 9 ⊢ Ⅎx(⟨A, B⟩Fy)
24	23	nfeq1 2499	. . . . . . . 8 ⊢ Ⅎx(⟨A, B⟩Fy) = z
25	19, 24	nfim 1813	. . . . . . 7 ⊢ Ⅎx(∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩Fy) = z)
26		nfv 1619	. . . . . . . 8 ⊢ Ⅎy∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R)
27		nfoprab2 5548	. . . . . . . . . . 11 ⊢ Ⅎy{⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
28	5, 27	nfcxfr 2487	. . . . . . . . . 10 ⊢ ℲyF
29	17, 28, 18	nfov 5546	. . . . . . . . 9 ⊢ Ⅎy(⟨A, B⟩F⟨C, D⟩)
30	29	nfeq1 2499	. . . . . . . 8 ⊢ Ⅎy(⟨A, B⟩F⟨C, D⟩) = z
31	26, 30	nfim 1813	. . . . . . 7 ⊢ Ⅎy(∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩F⟨C, D⟩) = z)
32		eqeq1 2359	. . . . . . . . . . 11 ⊢ (x = ⟨A, B⟩ → (x = ⟨w, v⟩ ↔ ⟨A, B⟩ = ⟨w, v⟩))
33	32	anbi1d 685	. . . . . . . . . 10 ⊢ (x = ⟨A, B⟩ → ((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩)))
34	33	anbi1d 685	. . . . . . . . 9 ⊢ (x = ⟨A, B⟩ → (((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)))
35	34	4exbidv 1630	. . . . . . . 8 ⊢ (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)))
36		oveq1 5531	. . . . . . . . 9 ⊢ (x = ⟨A, B⟩ → (xFy) = (⟨A, B⟩Fy))
37	36	eqeq1d 2361	. . . . . . . 8 ⊢ (x = ⟨A, B⟩ → ((xFy) = z ↔ (⟨A, B⟩Fy) = z))
38	35, 37	imbi12d 311	. . . . . . 7 ⊢ (x = ⟨A, B⟩ → ((∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) → (xFy) = z) ↔ (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩Fy) = z)))
39		eqeq1 2359	. . . . . . . . . . 11 ⊢ (y = ⟨C, D⟩ → (y = ⟨u, f⟩ ↔ ⟨C, D⟩ = ⟨u, f⟩))
40	39	anbi2d 684	. . . . . . . . . 10 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ↔ (⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩)))
41	40	anbi1d 685	. . . . . . . . 9 ⊢ (y = ⟨C, D⟩ → (((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R)))
42	41	4exbidv 1630	. . . . . . . 8 ⊢ (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)))
43		oveq2 5532	. . . . . . . . 9 ⊢ (y = ⟨C, D⟩ → (⟨A, B⟩Fy) = (⟨A, B⟩F⟨C, D⟩))
44	43	eqeq1d 2361	. . . . . . . 8 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩Fy) = z ↔ (⟨A, B⟩F⟨C, D⟩) = z))
45	42, 44	imbi12d 311	. . . . . . 7 ⊢ (y = ⟨C, D⟩ → ((∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩Fy) = z) ↔ (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩F⟨C, D⟩) = z)))
46		moeq 3013	. . . . . . . . . . . 12 ⊢ ∃*z z = R
47	46	mosubop 4614	. . . . . . . . . . 11 ⊢ ∃*z∃u∃f(y = ⟨u, f⟩ ∧ z = R)
48	47	mosubop 4614	. . . . . . . . . 10 ⊢ ∃*z∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R))
49		anass 630	. . . . . . . . . . . . . 14 ⊢ (((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
50	49	2exbii 1583	. . . . . . . . . . . . 13 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
51		19.42vv 1907	. . . . . . . . . . . . 13 ⊢ (∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
52	50, 51	bitri 240	. . . . . . . . . . . 12 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
53	52	2exbii 1583	. . . . . . . . . . 11 ⊢ (∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
54	53	mobii 2240	. . . . . . . . . 10 ⊢ (∃*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)))
55	48, 54	mpbir 200	. . . . . . . . 9 ⊢ ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)
56	55	a1i 10	. . . . . . . 8 ⊢ ((x ∈ (H × H) ∧ y ∈ (H × H)) → ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))
57	56, 5	ovidi 5595	. . . . . . 7 ⊢ ((x ∈ (H × H) ∧ y ∈ (H × H)) → (∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) → (xFy) = z))
58	16, 17, 18, 25, 31, 38, 45, 57	vtocl2gaf 2922	. . . . . 6 ⊢ ((⟨A, B⟩ ∈ (H × H) ∧ ⟨C, D⟩ ∈ (H × H)) → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩F⟨C, D⟩) = z))
59	14, 15, 58	syl2anbr 466	. . . . 5 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (∃w∃v∃u∃f((⟨A, B⟩ = ⟨w, v⟩ ∧ ⟨C, D⟩ = ⟨u, f⟩) ∧ z = R) → (⟨A, B⟩F⟨C, D⟩) = z))
60	13, 59	sylbird 226	. . . 4 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (z = S → (⟨A, B⟩F⟨C, D⟩) = z))
61		eqeq2 2362	. . . 4 ⊢ (z = S → ((⟨A, B⟩F⟨C, D⟩) = z ↔ (⟨A, B⟩F⟨C, D⟩) = S))
62	60, 61	mpbidi 207	. . 3 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (z = S → (⟨A, B⟩F⟨C, D⟩) = S))
63	3, 10, 62	exlimd 1806	. 2 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (∃z z = S → (⟨A, B⟩F⟨C, D⟩) = S))
64	2, 63	mpi 16	1 ⊢ (((A ∈ H ∧ B ∈ H) ∧ (C ∈ H ∧ D ∈ H)) → (⟨A, B⟩F⟨C, D⟩) = S)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2860  ⟨cop 4562   × cxp 4771  (class class class)co 5526  {coprab 5528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-ov 5527  df-oprab 5529

This theorem is used by: (None)