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Theorem f1ocnvfvb 5499
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 5498 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
213adant3 959 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
3 fveq2 5253 . . . . 5  |-  ( C  =  ( `' F `  D )  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
43eqcoms 2086 . . . 4  |-  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
5 f1ocnvfv2 5497 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( F `  ( `' F `  D ) )  =  D )
65eqeq2d 2094 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( F `  C )  =  ( F `  ( `' F `  D ) )  <->  ( F `  C )  =  D ) )
74, 6syl5ib 152 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
873adant2 958 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
92, 8impbid 127 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    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   `'ccnv 4400   -1-1-onto->wf1o 4968   ` cfv 4969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977
This theorem is referenced by:  f1ofveu  5579  f1ocnvfv3  5580
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