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 ) ) ) )