Theorem ov2gf 5888

Description: The value of an operation class abstraction. A version of ovmpog 5898 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 2692	. . 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 2290	. . . . 5 |- F/ x G e. _V
7		ov2gf.5	. . . . . . . 8 |- F = ( x e. C , y e. D |-> R )
8		nfmpo1 5831	. . . . . . . 8 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2276	. . . . . . 7 |- F/_ x F
10		nfcv 2279	. . . . . . 7 |- F/_ x y
11	2, 9, 10	nfov 5794	. . . . . 6 |- F/_ x ( A F y )
12	11, 5	nfeq 2287	. . . . 5 |- F/ x ( A F y ) = G
13	6, 12	nfim 1551	. . . 4 |- F/ x ( G e. _V -> ( A F y ) = G )
14		ov2gf.2	. . . . . 6 |- F/_ y S
15	14	nfel1 2290	. . . . 5 |- F/ y S e. _V
16		nfmpo2 5832	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2276	. . . . . . 7 |- F/_ y F
18	3, 17, 4	nfov 5794	. . . . . 6 |- F/_ y ( A F B )
19	18, 14	nfeq 2287	. . . . 5 |- F/ y ( A F B ) = S
20	15, 19	nfim 1551	. . . 4 |- F/ y ( S e. _V -> ( A F B ) = S )
21		ov2gf.3	. . . . . 6 |- ( x = A -> R = G )
22	21	eleq1d 2206	. . . . 5 |- ( x = A -> ( R e. _V <-> G e. _V ) )
23		oveq1 5774	. . . . . 6 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2152	. . . . 5 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = G ) )
25	22, 24	imbi12d 233	. . . 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 2206	. . . . 5 |- ( y = B -> ( G e. _V <-> S e. _V ) )
28		oveq2 5775	. . . . . 6 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2152	. . . . 5 |- ( y = B -> ( ( A F y ) = G <-> ( A F B ) = S ) )
30	27, 29	imbi12d 233	. . . 4 |- ( y = B -> ( ( G e. _V -> ( A F y ) = G ) <-> ( S e. _V -> ( A F B ) = S ) ) )
31	7	ovmpt4g 5886	. . . . 5 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1183	. . . 4 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2748	. . 3 |- ( ( A e. C /\ B e. D ) -> ( S e. _V -> ( A F B ) = S ) )
34	1, 33	syl5 32	. 2 |- ( ( A e. C /\ B e. D ) -> ( S e. H -> ( A F B ) = S ) )
35	34	3impia 1178	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   F/_wnfc 2266   _Vcvv 2681  (class class class)co 5767   e. cmpo 5769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772

This theorem is referenced by: (None)