Theorem ovmpt2x 5811

Description: The value of an operation class abstraction. Variant of ovmpt2ga 5812 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
ovmpt2x.1	|- ( ( x = A /\ y = B ) -> R = S )
ovmpt2x.2	|- ( x = A -> D = L )
ovmpt2x.3	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpt2x	|- ( ( 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 ovmpt2x

Step	Hyp	Ref	Expression
1		elex 2644	. 2 |- ( S e. H -> S e. _V )
2		ovmpt2x.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		ovmpt2x.1	. . . 4 |- ( ( x = A /\ y = B ) -> R = S )
5	4	adantl 272	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		ovmpt2x.2	. . . 4 |- ( x = A -> D = L )
7	6	adantl 272	. . 3 |- ( ( ( A e. C /\ B e. L /\ S e. _V ) /\ x = A ) -> D = L )
8		simp1 946	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> A e. C )
9		simp2 947	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> B e. L )
10		simp3 948	. . 3 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> S e. _V )
11	3, 5, 7, 8, 9, 10	ovmpt2dx 5809	. 2 |- ( ( A e. C /\ B e. L /\ S e. _V ) -> ( A F B ) = S )
12	1, 11	syl3an3 1216	1 |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 927   = wceq 1296   e. wcel 1445   _Vcvv 2633  (class class class)co 5690   |-> cmpt2 5692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695

This theorem is referenced by: (None)