Theorem f1ocpbllem 13392

Description: Lemma for f1ocpbl 13393. (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 5582	. . . . 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 1044	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> F : V -1-1-> X )
5		simp2l 1049	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
6		simp3l 1051	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
7		f1fveq 5912	. . 3 |- ( ( F : V -1-1-> X /\ ( A e. V /\ C e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
8	4, 5, 6, 7	syl12anc 1271	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
9		simp2r 1050	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10		simp3r 1052	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11		f1fveq 5912	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
12	4, 9, 10, 11	syl12anc 1271	. 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 1004   = wceq 1397   e. wcel 2202   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333  df-fv 5334

This theorem is referenced by:  f1ocpbl  13393  f1olecpbl  13395