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Theorem f1ocnvfvb 5991
Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
Assertion
Ref Expression
f1ocnvfvb  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  <-> 
( `' F `  D )  =  C ) )

Proof of Theorem f1ocnvfvb
StepHypRef Expression
1 f1ocnvfv 5990 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
213adant3 1008 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
3 fveq2 5696 . . . . 5  |-  ( C  =  ( `' F `  D )  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
43eqcoms 2446 . . . 4  |-  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
5 f1ocnvfv2 5989 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( F `  ( `' F `  D ) )  =  D )
65eqeq2d 2454 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( F `  C )  =  ( F `  ( `' F `  D ) )  <->  ( F `  C )  =  D ) )
74, 6syl5ib 219 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
873adant2 1007 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
92, 8impbid 191 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  <-> 
( `' F `  D )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   `'ccnv 4844   -1-1-onto->wf1o 5422   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431
This theorem is referenced by:  f1ofveu  6091  f1ocnvfv3  6092  1arith2  13994  f1omvdcnv  15955  f1omvdconj  15957  txhmeo  19381  iccpnfcnv  20521  dvcnvlem  21453  logeftb  22037  sqff1o  22525  bracnlnval  25523  cdlemg17h  34317
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