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Metamath Proof Explorer

Theorem ovmpt4d

Description: Deduction version of ovmpt4g . (This is the operation analogue of fvmpt2d .) (Contributed by Zhi Wang, 9-Oct-2025)

Ref Expression
Hypotheses ovmpt4d.1 
|- ( ph -> F = ( x e. A , y e. B |-> C ) )
ovmpt4d.2 
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. V )
Assertion ovmpt4d 
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = C )

Proof

Step Hyp Ref Expression
1 ovmpt4d.1 
 |-  ( ph -> F = ( x e. A , y e. B |-> C ) )
2 ovmpt4d.2 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. V )
3 1 oveqdr 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = ( x ( x e. A , y e. B |-> C ) y ) )
4 simprl 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> x e. A )
5 simprr 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> y e. B )
6 eqid 
 |-  ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> C )
7 6 ovmpt4g 
 |-  ( ( x e. A /\ y e. B /\ C e. V ) -> ( x ( x e. A , y e. B |-> C ) y ) = C )
8 4 5 2 7 syl3anc 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x ( x e. A , y e. B |-> C ) y ) = C )
9 3 8 eqtrd 
 |-  ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = C )