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Theorem f1orescnv 5651
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 5648 . . 3  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
21adantl 466 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
3 funcnvres 5482 . . . 4  |-  ( Fun  `' F  ->  `' ( F  |`  R )  =  ( `' F  |`  ( F " R
) ) )
4 df-ima 4848 . . . . . 6  |-  ( F
" R )  =  ran  ( F  |`  R )
5 dff1o5 5645 . . . . . . 7  |-  ( ( F  |`  R ) : R -1-1-onto-> P  <->  ( ( F  |`  R ) : R -1-1-> P  /\  ran  ( F  |`  R )  =  P ) )
65simprbi 464 . . . . . 6  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ran  ( F  |`  R )  =  P )
74, 6syl5eq 2482 . . . . 5  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( F " R )  =  P )
87reseq2d 5105 . . . 4  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( `' F  |`  ( F " R ) )  =  ( `' F  |`  P ) )
93, 8sylan9eq 2490 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R )  =  ( `' F  |`  P ) )
10 f1oeq1 5627 . . 3  |-  ( `' ( F  |`  R )  =  ( `' F  |`  P )  ->  ( `' ( F  |`  R ) : P -1-1-onto-> R  <->  ( `' F  |`  P ) : P -1-1-onto-> R ) )
119, 10syl 16 . 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 210 1  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  ( `' F  |`  P ) : P -1-1-onto-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   `'ccnv 4834   ran crn 4836    |` cres 4837   "cima 4838   Fun wfun 5407   -1-1->wf1 5410   -1-1-onto->wf1o 5412
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
This theorem is referenced by:  f1oresrab  5870  relogf1o  21993
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