Metamath Proof Explorer


Theorem f1olecpbl

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

Ref Expression
Hypothesis f1ocpbl.f
|- ( ph -> F : V -1-1-onto-> X )
Assertion f1olecpbl
|- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N 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 breq12
 |-  ( ( A = C /\ B = D ) -> ( A N B <-> C N D ) )
4 2 3 syl6bi
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )