Theorem ov2gf 6785

Description: The value of an operation class abstraction. A version of ovmpt2g 6795 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 3212	. . 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 2779	. . . . 5 |- F/ x G e. _V
7		ov2gf.5	. . . . . . . 8 |- F = ( x e. C , y e. D |-> R )
8		nfmpt21 6722	. . . . . . . 8 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2762	. . . . . . 7 |- F/_ x F
10		nfcv 2764	. . . . . . 7 |- F/_ x y
11	2, 9, 10	nfov 6676	. . . . . 6 |- F/_ x ( A F y )
12	11, 5	nfeq 2776	. . . . 5 |- F/ x ( A F y ) = G
13	6, 12	nfim 1825	. . . 4 |- F/ x ( G e. _V -> ( A F y ) = G )
14		ov2gf.2	. . . . . 6 |- F/_ y S
15	14	nfel1 2779	. . . . 5 |- F/ y S e. _V
16		nfmpt22 6723	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2762	. . . . . . 7 |- F/_ y F
18	3, 17, 4	nfov 6676	. . . . . 6 |- F/_ y ( A F B )
19	18, 14	nfeq 2776	. . . . 5 |- F/ y ( A F B ) = S
20	15, 19	nfim 1825	. . . 4 |- F/ y ( S e. _V -> ( A F B ) = S )
21		ov2gf.3	. . . . . 6 |- ( x = A -> R = G )
22	21	eleq1d 2686	. . . . 5 |- ( x = A -> ( R e. _V <-> G e. _V ) )
23		oveq1 6657	. . . . . 6 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = G ) )
25	22, 24	imbi12d 334	. . . 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 2686	. . . . 5 |- ( y = B -> ( G e. _V <-> S e. _V ) )
28		oveq2 6658	. . . . . 6 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2637	. . . . 5 |- ( y = B -> ( ( A F y ) = G <-> ( A F B ) = S ) )
30	27, 29	imbi12d 334	. . . 4 |- ( y = B -> ( ( G e. _V -> ( A F y ) = G ) <-> ( S e. _V -> ( A F B ) = S ) ) )
31	7	ovmpt4g 6783	. . . . 5 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1267	. . . 4 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 3273	. . 3 |- ( ( A e. C /\ B e. D ) -> ( S e. _V -> ( A F B ) = S ) )
34	1, 33	syl5 34	. 2 |- ( ( A e. C /\ B e. D ) -> ( S e. H -> ( A F B ) = S ) )
35	34	3impia 1261	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by: (None)