Theorem ov2gf 5711

Description: The value of an operation class abstraction. A version of ovmpt2g 5715 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ov2gf.a	⊢ ℲxA
ov2gf.c	⊢ ℲyA
ov2gf.d	⊢ ℲyB
ov2gf.1	⊢ ℲxG
ov2gf.2	⊢ ℲyS
ov2gf.3	⊢ (x = A → R = G)
ov2gf.4	⊢ (y = B → G = S)
ov2gf.5	⊢ F = (x ∈ C, y ∈ D ↦ R)

Assertion

Ref	Expression
ov2gf	⊢ ((A ∈ C ∧ B ∈ D ∧ S ∈ H) → (AFB) = S)

Distinct variable groups:   x,y,C   x,D,y

Allowed substitution hints:   A(x,y)   B(x,y)   R(x,y)   S(x,y)   F(x,y)   G(x,y)   H(x,y)

Proof of Theorem ov2gf

Step	Hyp	Ref	Expression
1		elex 2867	. . 3 ⊢ (S ∈ H → S ∈ V)
2		ov2gf.a	. . . 4 ⊢ ℲxA
3		ov2gf.c	. . . 4 ⊢ ℲyA
4		ov2gf.d	. . . 4 ⊢ ℲyB
5		ov2gf.1	. . . . . 6 ⊢ ℲxG
6	5	nfel1 2499	. . . . 5 ⊢ Ⅎx G ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ F = (x ∈ C, y ∈ D ↦ R)
8		nfmpt21 5673	. . . . . . . 8 ⊢ Ⅎx(x ∈ C, y ∈ D ↦ R)
9	7, 8	nfcxfr 2486	. . . . . . 7 ⊢ ℲxF
10		nfcv 2489	. . . . . . 7 ⊢ Ⅎxy
11	2, 9, 10	nfov 5545	. . . . . 6 ⊢ Ⅎx(AFy)
12	11, 5	nfeq 2496	. . . . 5 ⊢ Ⅎx(AFy) = G
13	6, 12	nfim 1813	. . . 4 ⊢ Ⅎx(G ∈ V → (AFy) = G)
14		ov2gf.2	. . . . . 6 ⊢ ℲyS
15	14	nfel1 2499	. . . . 5 ⊢ Ⅎy S ∈ V
16		nfmpt22 5674	. . . . . . . 8 ⊢ Ⅎy(x ∈ C, y ∈ D ↦ R)
17	7, 16	nfcxfr 2486	. . . . . . 7 ⊢ ℲyF
18	3, 17, 4	nfov 5545	. . . . . 6 ⊢ Ⅎy(AFB)
19	18, 14	nfeq 2496	. . . . 5 ⊢ Ⅎy(AFB) = S
20	15, 19	nfim 1813	. . . 4 ⊢ Ⅎy(S ∈ V → (AFB) = S)
21		ov2gf.3	. . . . . 6 ⊢ (x = A → R = G)
22	21	eleq1d 2419	. . . . 5 ⊢ (x = A → (R ∈ V ↔ G ∈ V))
23		oveq1 5530	. . . . . 6 ⊢ (x = A → (xFy) = (AFy))
24	23, 21	eqeq12d 2367	. . . . 5 ⊢ (x = A → ((xFy) = R ↔ (AFy) = G))
25	22, 24	imbi12d 311	. . . 4 ⊢ (x = A → ((R ∈ V → (xFy) = R) ↔ (G ∈ V → (AFy) = G)))
26		ov2gf.4	. . . . . 6 ⊢ (y = B → G = S)
27	26	eleq1d 2419	. . . . 5 ⊢ (y = B → (G ∈ V ↔ S ∈ V))
28		oveq2 5531	. . . . . 6 ⊢ (y = B → (AFy) = (AFB))
29	28, 26	eqeq12d 2367	. . . . 5 ⊢ (y = B → ((AFy) = G ↔ (AFB) = S))
30	27, 29	imbi12d 311	. . . 4 ⊢ (y = B → ((G ∈ V → (AFy) = G) ↔ (S ∈ V → (AFB) = S)))
31	7	ovmpt4g 5710	. . . . 5 ⊢ ((x ∈ C ∧ y ∈ D ∧ R ∈ V) → (xFy) = R)
32	31	3expia 1153	. . . 4 ⊢ ((x ∈ C ∧ y ∈ D) → (R ∈ V → (xFy) = R))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2921	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (S ∈ V → (AFB) = S))
34	1, 33	syl5 28	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (S ∈ H → (AFB) = S))
35	34	3impia 1148	1 ⊢ ((A ∈ C ∧ B ∈ D ∧ S ∈ H) → (AFB) = S)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  (class class class)co 5525   ↦ cmpt2 5653

This theorem was proved from 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528  df-mpt2 5654

This theorem is referenced by: (None)