Theorem ov2gf 5712

Description: The value of an operation class abstraction. A version of ovmpt2g 5716 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 2868	. . 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 2500	. . . . 5 ⊢ Ⅎx G ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ F = (x ∈ C, y ∈ D ↦ R)
8		nfmpt21 5674	. . . . . . . 8 ⊢ Ⅎx(x ∈ C, y ∈ D ↦ R)
9	7, 8	nfcxfr 2487	. . . . . . 7 ⊢ ℲxF
10		nfcv 2490	. . . . . . 7 ⊢ Ⅎxy
11	2, 9, 10	nfov 5546	. . . . . 6 ⊢ Ⅎx(AFy)
12	11, 5	nfeq 2497	. . . . 5 ⊢ Ⅎx(AFy) = G
13	6, 12	nfim 1813	. . . 4 ⊢ Ⅎx(G ∈ V → (AFy) = G)
14		ov2gf.2	. . . . . 6 ⊢ ℲyS
15	14	nfel1 2500	. . . . 5 ⊢ Ⅎy S ∈ V
16		nfmpt22 5675	. . . . . . . 8 ⊢ Ⅎy(x ∈ C, y ∈ D ↦ R)
17	7, 16	nfcxfr 2487	. . . . . . 7 ⊢ ℲyF
18	3, 17, 4	nfov 5546	. . . . . 6 ⊢ Ⅎy(AFB)
19	18, 14	nfeq 2497	. . . . 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 5531	. . . . . 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 5532	. . . . . 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 5711	. . . . 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 2922	. . 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 2477  Vcvv 2860  (class class class)co 5526   ↦ cmpt2 5654

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 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 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 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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-ov 5527  df-oprab 5529  df-mpt2 5655

This theorem is referenced by: (None)