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Metamath Proof Explorer

Theorem ovmpox

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

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

Proof

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