ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1orescnv Unicode version

Theorem f1orescnv 5430
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 5427 . . 3  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
21adantl 275 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
3 funcnvres 5243 . . . 4  |-  ( Fun  `' F  ->  `' ( F  |`  R )  =  ( `' F  |`  ( F " R
) ) )
4 df-ima 4599 . . . . . 6  |-  ( F
" R )  =  ran  ( F  |`  R )
5 dff1o5 5423 . . . . . . 7  |-  ( ( F  |`  R ) : R -1-1-onto-> P  <->  ( ( F  |`  R ) : R -1-1-> P  /\  ran  ( F  |`  R )  =  P ) )
65simprbi 273 . . . . . 6  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ran  ( F  |`  R )  =  P )
74, 6syl5eq 2202 . . . . 5  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( F " R )  =  P )
87reseq2d 4866 . . . 4  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( `' F  |`  ( F " R ) )  =  ( `' F  |`  P ) )
93, 8sylan9eq 2210 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R )  =  ( `' F  |`  P ) )
10 f1oeq1 5403 . . 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 1335   `'ccnv 4585   ran crn 4587    |` cres 4588   "cima 4589   Fun wfun 5164   -1-1->wf1 5167   -1-1-onto->wf1o 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177
This theorem is referenced by:  f1oresrab  5632
  Copyright terms: Public domain W3C validator