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Theorem f1ocpbllem 13749
Description: Lemma for f1ocpbl 13750. (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
StepHypRef Expression
1 f1ocpbl.f . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> X )
2 f1of1 5673 . . . . 5  |-  ( F : V -1-1-onto-> X  ->  F : V -1-1-> X )
31, 2syl 16 . . . 4  |-  ( ph  ->  F : V -1-1-> X
)
433ad2ant1 978 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  F : V -1-1-> X )
5 simp2l 983 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
6 simp3l 985 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  C  e.  V )
7 f1fveq 6008 . . 3  |-  ( ( F : V -1-1-> X  /\  ( A  e.  V  /\  C  e.  V
) )  ->  (
( F `  A
)  =  ( F `
 C )  <->  A  =  C ) )
84, 5, 6, 7syl12anc 1182 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  =  C ) )
9 simp2r 984 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
10 simp3r 986 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  D  e.  V )
11 f1fveq 6008 . . 3  |-  ( ( F : V -1-1-> X  /\  ( B  e.  V  /\  D  e.  V
) )  ->  (
( F `  B
)  =  ( F `
 D )  <->  B  =  D ) )
124, 9, 10, 11syl12anc 1182 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  =  D ) )
138, 12anbi12d 692 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 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454
This theorem is referenced by:  f1ocpbl  13750  f1olecpbl  13752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-f1o 5461  df-fv 5462
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