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Theorem f1ocnvfvb 6186
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 6185 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
213adant3 1016 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
3 fveq2 5872 . . . . 5  |-  ( C  =  ( `' F `  D )  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
43eqcoms 2469 . . . 4  |-  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
5 f1ocnvfv2 6184 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( F `  ( `' F `  D ) )  =  D )
65eqeq2d 2471 . . . 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 1015 . 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 973    = wceq 1395    e. wcel 1819   `'ccnv 5007   -1-1-onto->wf1o 5593   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  f1ofveu  6291  f1ocnvfv3  6292  1arith2  14458  f1omvdcnv  16596  f1omvdconj  16598  txhmeo  20430  iccpnfcnv  21570  dvcnvlem  22503  logeftb  23094  sqff1o  23582  bracnlnval  27160  cdlemg17h  36537
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