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Theorem f1ocpbllem 13442
Description: Lemma for f1ocpbl 13443. (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 5487 . . . . 5  |-  ( F : V -1-1-onto-> X  ->  F : V -1-1-> X )
31, 2syl 15 . . . 4  |-  ( ph  ->  F : V -1-1-> X
)
433ad2ant1 976 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  F : V -1-1-> X )
5 simp2l 981 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
6 simp3l 983 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  C  e.  V )
7 f1fveq 5802 . . 3  |-  ( ( F : V -1-1-> X  /\  ( A  e.  V  /\  C  e.  V
) )  ->  (
( F `  A
)  =  ( F `
 C )  <->  A  =  C ) )
84, 5, 6, 7syl12anc 1180 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  =  C ) )
9 simp2r 982 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
10 simp3r 984 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  D  e.  V )
11 f1fveq 5802 . . 3  |-  ( ( F : V -1-1-> X  /\  ( B  e.  V  /\  D  e.  V
) )  ->  (
( F `  B
)  =  ( F `
 D )  <->  B  =  D ) )
124, 9, 10, 11syl12anc 1180 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  =  D ) )
138, 12anbi12d 691 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem is referenced by:  f1ocpbl  13443  f1olecpbl  13445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279
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