Theorem ovmpox 6047

Description: The value of an operation class abstraction. Variant of ovmpoga 6048 which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Hypotheses

Ref	Expression
ovmpox.1	|- ( ( x = A /\ y = B ) -> R = S )
ovmpox.2	|- ( x = A -> D = L )
ovmpox.3	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpox	|- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, L, y   x, S, y

Allowed substitution hints:   D( x, y)   R( x, y)   F( x, y)   H( x, y)

Proof of Theorem ovmpox

Step	Hyp	Ref	Expression
1		elex 2771	. 2 |- ( S e. H -> S e. _V )
2		ovmpox.3	. . . 4 |- F = ( x e. C , y e. D |-> R )
3	2	a1i 9	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> F = ( x e. C , y e. D |-> R ) )
4		ovmpox.1	. . . 4 |- ( ( x = A /\ y = B ) -> R = S )
5	4	adantl 277	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		ovmpox.2	. . . 4 |- ( x = A -> D = L )
7	6	adantl 277	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ x = A ) -> D = L )
8		simp1 999	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> A e. C )
9		simp2 1000	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> B e. L )
10		simp3 1001	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> S e. _V )
11	3, 5, 7, 8, 9, 10	ovmpodx 6045	. 2 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> ( A F B ) = S )
12	1, 11	syl3an3 1284	1 |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   _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:  reldvg  14833