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Metamath Proof Explorer

Theorem f1ocpbllem

Description: Lemma for f1ocpbl . (Contributed by Mario Carneiro, 24-Feb-2015)

Ref Expression
Hypothesis f1ocpbl.f 
|- ( ph -> F : V -1-1-onto-> X )
Assertion 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 ) ) )

Proof

Step Hyp Ref Expression
1 f1ocpbl.f 
 |-  ( ph -> F : V -1-1-onto-> X )
2 f1of1 
 |-  ( F : V -1-1-onto-> X -> F : V -1-1-> X )
3 1 2 syl 
 |-  ( ph -> F : V -1-1-> X )
4 3 3ad2ant1 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> F : V -1-1-> X )
5 simp2l 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
6 simp3l 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
7 f1fveq 
 |-  ( ( F : V -1-1-> X /\ ( A e. V /\ C e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
8 4 5 6 7 syl12anc 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
9 simp2r 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10 simp3r 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11 f1fveq 
 |-  ( ( F : V -1-1-> X /\ ( B e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
12 4 9 10 11 syl12anc 
 |-  ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
13 8 12 anbi12d 
 |-  ( ( 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 ) ) )