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Theorem f1ocnvfvb 6020
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 6019 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
213adant3 978 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
3 fveq2 5731 . . . . 5  |-  ( C  =  ( `' F `  D )  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
43eqcoms 2441 . . . 4  |-  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
5 f1ocnvfv2 6018 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( F `  ( `' F `  D ) )  =  D )
65eqeq2d 2449 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( F `  C )  =  ( F `  ( `' F `  D ) )  <->  ( F `  C )  =  D ) )
74, 6syl5ib 212 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
873adant2 977 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
92, 8impbid 185 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  <-> 
( `' F `  D )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   `'ccnv 4880   -1-1-onto->wf1o 5456   ` cfv 5457
This theorem is referenced by:  f1ofveu  6587  f1ocnvfv3  6588  1arith2  13301  txhmeo  17840  iccpnfcnv  18974  dvcnvlem  19865  logeftb  20483  sqff1o  20970  bracnlnval  23622  f1omvdcnv  27378  f1omvdconj  27380  cdlemg17h  31539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465
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