Theorem ov2gf 5977

Description: The value of an operation class abstraction. A version of ovmpog 5987 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 2741	. . 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 2323	. . . . 5 |- F/ x G e. _V
7		ov2gf.5	. . . . . . . 8 |- F = ( x e. C , y e. D |-> R )
8		nfmpo1 5920	. . . . . . . 8 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2309	. . . . . . 7 |- F/_ x F
10		nfcv 2312	. . . . . . 7 |- F/_ x y
11	2, 9, 10	nfov 5883	. . . . . 6 |- F/_ x ( A F y )
12	11, 5	nfeq 2320	. . . . 5 |- F/ x ( A F y ) = G
13	6, 12	nfim 1565	. . . 4 |- F/ x ( G e. _V -> ( A F y ) = G )
14		ov2gf.2	. . . . . 6 |- F/_ y S
15	14	nfel1 2323	. . . . 5 |- F/ y S e. _V
16		nfmpo2 5921	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2309	. . . . . . 7 |- F/_ y F
18	3, 17, 4	nfov 5883	. . . . . 6 |- F/_ y ( A F B )
19	18, 14	nfeq 2320	. . . . 5 |- F/ y ( A F B ) = S
20	15, 19	nfim 1565	. . . 4 |- F/ y ( S e. _V -> ( A F B ) = S )
21		ov2gf.3	. . . . . 6 |- ( x = A -> R = G )
22	21	eleq1d 2239	. . . . 5 |- ( x = A -> ( R e. _V <-> G e. _V ) )
23		oveq1 5860	. . . . . 6 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2185	. . . . 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 2239	. . . . 5 |- ( y = B -> ( G e. _V <-> S e. _V ) )
28		oveq2 5861	. . . . . 6 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2185	. . . . 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 5975	. . . . 5 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1200	. . . 4 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2797	. . 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 1195	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730  (class class class)co 5853   e. cmpo 5855

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858

This theorem is referenced by: (None)