Theorem f1ocpbllem 13142

Description: Lemma for f1ocpbl 13143. (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 5521	. . . . 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 1021	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> F : V -1-1-> X )
5		simp2l 1026	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
6		simp3l 1028	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> C e. V )
7		f1fveq 5841	. . 3 |- ( ( F : V -1-1-> X /\ ( A e. V /\ C e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
8	4, 5, 6, 7	syl12anc 1248	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
9		simp2r 1027	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
10		simp3r 1029	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> D e. V )
11		f1fveq 5841	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B = D ) )
12	4, 9, 10, 11	syl12anc 1248	. 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 981   = wceq 1373   e. wcel 2176   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279

This theorem is referenced by:  f1ocpbl  13143  f1olecpbl  13145