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 ) ) ) |
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 ) ) ) |