Theorem ovmpox 6182

Description: The value of an operation class abstraction. Variant of ovmpoga 6183 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 2825	. 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 1024	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> A e. C )
9		simp2 1025	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> B e. L )
10		simp3 1026	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> S e. _V )
11	3, 5, 7, 8, 9, 10	ovmpodx 6180	. 2 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> ( A F B ) = S )
12	1, 11	syl3an3 1309	1 |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   _Vcvv 2813  (class class class)co 6050   e. cmpo 6052

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055

This theorem is referenced by:  reldvg  15544