MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1orescnv Unicode version

Theorem f1orescnv 5657
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 5654 . . 3  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
21adantl 453 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R ) : P -1-1-onto-> R )
3 funcnvres 5489 . . . 4  |-  ( Fun  `' F  ->  `' ( F  |`  R )  =  ( `' F  |`  ( F " R
) ) )
4 df-ima 4858 . . . . . 6  |-  ( F
" R )  =  ran  ( F  |`  R )
5 dff1o5 5650 . . . . . . 7  |-  ( ( F  |`  R ) : R -1-1-onto-> P  <->  ( ( F  |`  R ) : R -1-1-> P  /\  ran  ( F  |`  R )  =  P ) )
65simprbi 451 . . . . . 6  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ran  ( F  |`  R )  =  P )
74, 6syl5eq 2456 . . . . 5  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( F " R )  =  P )
87reseq2d 5113 . . . 4  |-  ( ( F  |`  R ) : R -1-1-onto-> P  ->  ( `' F  |`  ( F " R ) )  =  ( `' F  |`  P ) )
93, 8sylan9eq 2464 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  `' ( F  |`  R )  =  ( `' F  |`  P ) )
10 f1oeq1 5632 . . 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 202 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    <-> wb 177    /\ wa 359    = wceq 1649   `'ccnv 4844   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5415   -1-1->wf1 5418   -1-1-onto->wf1o 5420
This theorem is referenced by:  relogf1o  20425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428
  Copyright terms: Public domain W3C validator