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Theorem f1orescnv 5284
Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
Assertion
Ref Expression
f1orescnv  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  ( `' F  |`  P ) : P -1-1-onto-> R )

Proof of Theorem f1orescnv
StepHypRef Expression
1 f1ocnv 5281 . . 3  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
21adantl 272 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
3 funcnvres 5102 . . . 4  |-  ( Fun  `' F  ->  `' ( F  |`  R )  =  ( `' F  |`  ( F " R
) ) )
4 df-ima 4467 . . . . . 6  |-  ( F
" R )  =  ran  ( F  |`  R )
5 dff1o5 5277 . . . . . . 7  |-  ( ( F  |`  R ) : R -1-1-onto-> P  <->  ( ( F  |`  R ) : R -1-1-> P  /\  ran  ( F  |`  R )  =  P ) )
65simprbi 270 . . . . . 6  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ran  ( F  |`  R )  =  P )
74, 6syl5eq 2133 . . . . 5  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( F " R )  =  P )
87reseq2d 4728 . . . 4  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( `' F  |`  ( F " R ) )  =  ( `' F  |`  P ) )
93, 8sylan9eq 2141 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R )  =  ( `' F  |`  P ) )
10 f1oeq1 5259 . . 3  |-  ( `' ( F  |`  R )  =  ( `' F  |`  P )  ->  ( `' ( F  |`  R ) : P -1-1-onto-> R  <->  ( `' F  |`  P ) : P -1-1-onto-> R ) )
119, 10syl 14 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  ( `' ( F  |`  R ) : P -1-1-onto-> R  <->  ( `' F  |`  P ) : P -1-1-onto-> R ) )
122, 11mpbid 146 1  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  ( `' F  |`  P ) : P -1-1-onto-> R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290   `'ccnv 4453   ran crn 4455    |` cres 4456   "cima 4457   Fun wfun 5024   -1-1->wf1 5027   -1-1-onto->wf1o 5029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037
This theorem is referenced by:  f1oresrab  5479
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