Theorem f1ocpbllem 13227

Description: Lemma for f1ocpbl 13228. (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 5538	. . . . 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 5859	. . 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 5859	. . 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 2177   -1-1->wf1 5282   -1-1-onto->wf1o 5284   ` cfv 5285

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-f1o 5292  df-fv 5293

This theorem is referenced by:  f1ocpbl  13228  f1olecpbl  13230