Theorem ov2gf 6043

Description: The value of an operation class abstraction. A version of ovmpog 6053 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 2771	. . 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 2347	. . . . 5 |- F/ x G e. _V
7		ov2gf.5	. . . . . . . 8 |- F = ( x e. C , y e. D |-> R )
8		nfmpo1 5985	. . . . . . . 8 |- F/_ x ( x e. C , y e. D |-> R )
9	7, 8	nfcxfr 2333	. . . . . . 7 |- F/_ x F
10		nfcv 2336	. . . . . . 7 |- F/_ x y
11	2, 9, 10	nfov 5948	. . . . . 6 |- F/_ x ( A F y )
12	11, 5	nfeq 2344	. . . . 5 |- F/ x ( A F y ) = G
13	6, 12	nfim 1583	. . . 4 |- F/ x ( G e. _V -> ( A F y ) = G )
14		ov2gf.2	. . . . . 6 |- F/_ y S
15	14	nfel1 2347	. . . . 5 |- F/ y S e. _V
16		nfmpo2 5986	. . . . . . . 8 |- F/_ y ( x e. C , y e. D |-> R )
17	7, 16	nfcxfr 2333	. . . . . . 7 |- F/_ y F
18	3, 17, 4	nfov 5948	. . . . . 6 |- F/_ y ( A F B )
19	18, 14	nfeq 2344	. . . . 5 |- F/ y ( A F B ) = S
20	15, 19	nfim 1583	. . . 4 |- F/ y ( S e. _V -> ( A F B ) = S )
21		ov2gf.3	. . . . . 6 |- ( x = A -> R = G )
22	21	eleq1d 2262	. . . . 5 |- ( x = A -> ( R e. _V <-> G e. _V ) )
23		oveq1 5925	. . . . . 6 |- ( x = A -> ( x F y ) = ( A F y ) )
24	23, 21	eqeq12d 2208	. . . . 5 |- ( x = A -> ( ( x F y ) = R <-> ( A F y ) = G ) )
25	22, 24	imbi12d 234	. . . 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 2262	. . . . 5 |- ( y = B -> ( G e. _V <-> S e. _V ) )
28		oveq2 5926	. . . . . 6 |- ( y = B -> ( A F y ) = ( A F B ) )
29	28, 26	eqeq12d 2208	. . . . 5 |- ( y = B -> ( ( A F y ) = G <-> ( A F B ) = S ) )
30	27, 29	imbi12d 234	. . . 4 |- ( y = B -> ( ( G e. _V -> ( A F y ) = G ) <-> ( S e. _V -> ( A F B ) = S ) ) )
31	7	ovmpt4g 6041	. . . . 5 |- ( ( x e. C /\ y e. D /\ R e. _V ) -> ( x F y ) = R )
32	31	3expia 1207	. . . 4 |- ( ( x e. C /\ y e. D ) -> ( R e. _V -> ( x F y ) = R ) )
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2827	. . 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 1202	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   F/_wnfc 2323   _Vcvv 2760  (class class class)co 5918   e. cmpo 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923

This theorem is referenced by: (None)