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Metamath Proof Explorer

Theorem f1ocpbl

Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015)

Ref Expression
Hypothesis f1ocpbl.f 
|- ( ph -> F : V -1-1-onto-> X )
Assertion f1ocpbl 
|- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )

Proof

Step Hyp Ref Expression
1 f1ocpbl.f 
 |-  ( ph -> F : V -1-1-onto-> X )
2 1 f1ocpbllem 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) <-> ( A = C /\ B = D ) ) )
3 oveq12 
 |-  ( ( A = C /\ B = D ) -> ( A .+ B ) = ( C .+ D ) )
4 3 fveq2d 
 |-  ( ( A = C /\ B = D ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) )
5 2 4 biimtrdi 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )