Theorem f1ocpbllem 13113

Description: Lemma for f1ocpbl 13114. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
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 of Theorem f1ocpbllem

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . . . 5 |- ( ph -> F : V -1-1-onto-> X )
2		f1of1 5520	. . . . 5 |- ( F : V -1-1-onto-> X -> F : V -1-1-> X )
3	1, 2	syl 14	. . . 4 |- ( ph -> F : V -1-1-> X )
4	3	3ad2ant1 1020	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> F : V -1-1-> X )
5		simp2l 1025	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
6		simp3l 1027	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
7		f1fveq 5840	. . 3 |- ( ( F : V -1-1-> X /\ ( A e. V /\ C e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
8	4, 5, 6, 7	syl12anc 1247	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
9		simp2r 1026	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10		simp3r 1028	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11		f1fveq 5840	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
12	4, 9, 10, 11	syl12anc 1247	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
13	8, 12	anbi12d 473	1 |- ( ( 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   -1-1->wf1 5267   -1-1-onto->wf1o 5269   ` cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-f1o 5277  df-fv 5278

This theorem is referenced by:  f1ocpbl  13114  f1olecpbl  13116