Theorem ov2gf 5625

Description: The value of an operation class abstraction. A version of ovmpt2g 5635 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ov2gf.a	|- F/_ x A
ov2gf.c	|- F/_ y A
ov2gf.d	|- F/_ y B
ov2gf.1	|- F/_ x G
ov2gf.2	|- F/_ y S
ov2gf.3	|- ( x = A -> R = G )
ov2gf.4	|- ( y = B -> G = S )
ov2gf.5	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ov2gf	|- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = 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 2566	. . 3 |- ( S e. H -> S e. _V )
2		ov2gf.a	. . . 4 |- F/_ x A
3		ov2gf.c	. . . 4 |- F/_ y A
4		ov2gf.d	. . . 4 |- F/_ y B
5		ov2gf.1	. . . . . 6 |- F/_ x G
6	5	nfel1 2188	. . . . 5 |- F/ x G e. _V
7		ov2gf.5	. . . . . . . 8 |- F = ( x e. C , y e. D |-> R )
8		nfmpt21 5571	. . . . . . . 8 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2175	. . . . . . 7 |- F/_ x F
10		nfcv 2178	. . . . . . 7 |- F/_ x y
11	2, 9, 10	nfov 5535	. . . . . 6 |- F/_ x ( A F y )
12	11, 5	nfeq 2185	. . . . 5 |- F/ x ( A F y ) = G
13	6, 12	nfim 1464	. . . 4 |- F/ x ( G e. _V -> ( A F y ) = G )
14		ov2gf.2	. . . . . 6 |- F/_ y S
15	14	nfel1 2188	. . . . 5 |- F/ y S e. _V
16		nfmpt22 5572	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2175	. . . . . . 7 |- F/_ y F
18	3, 17, 4	nfov 5535	. . . . . 6 |- F/_ y ( A F B )
19	18, 14	nfeq 2185	. . . . 5 |- F/ y ( A F B ) = S
20	15, 19	nfim 1464	. . . 4 |- F/ y ( S e. _V -> ( A F B ) = S )
21		ov2gf.3	. . . . . 6 |- ( x = A -> R = G )
22	21	eleq1d 2106	. . . . 5 |- ( x = A -> ( R e. _V <-> G e. _V ) )
23		oveq1 5519	. . . . . 6 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2054	. . . . 5 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = G ) )
25	22, 24	imbi12d 223	. . . 4 |- ( x = A -> ( ( R e. _V -> ( x F y ) = R ) <-> ( G e. _V -> ( A F y ) = G ) ) )
26		ov2gf.4	. . . . . 6 |- ( y = B -> G = S )
27	26	eleq1d 2106	. . . . 5 |- ( y = B -> ( G e. _V <-> S e. _V ) )
28		oveq2 5520	. . . . . 6 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2054	. . . . 5 |- ( y = B -> ( ( A F y ) = G <-> ( A F B ) = S ) )
30	27, 29	imbi12d 223	. . . 4 |- ( y = B -> ( ( G e. _V -> ( A F y ) = G ) <-> ( S e. _V -> ( A F B ) = S ) ) )
31	7	ovmpt4g 5623	. . . . 5 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1106	. . . 4 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2620	. . 3 |- ( ( A e. C /\ B e. D ) -> ( S e. _V -> ( A F B ) = S ) )
34	1, 33	syl5 28	. 2 |- ( ( A e. C /\ B e. D ) -> ( S e. H -> ( A F B ) = S ) )
35	34	3impia 1101	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   F/_wnfc 2165   _Vcvv 2557  (class class class)co 5512   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)