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

Theorem funcnvres 5166
Description: The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
funcnvres  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )

Proof of Theorem funcnvres
StepHypRef Expression
1 df-ima 4522 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 df-rn 4520 . . . 4  |-  ran  ( F  |`  A )  =  dom  `' ( F  |`  A )
31, 2eqtri 2138 . . 3  |-  ( F
" A )  =  dom  `' ( F  |`  A )
43reseq2i 4786 . 2  |-  ( `' F  |`  ( F " A ) )  =  ( `' F  |`  dom  `' ( F  |`  A ) )
5 resss 4813 . . . 4  |-  ( F  |`  A )  C_  F
6 cnvss 4682 . . . 4  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
75, 6ax-mp 5 . . 3  |-  `' ( F  |`  A )  C_  `' F
8 funssres 5135 . . 3  |-  ( ( Fun  `' F  /\  `' ( F  |`  A )  C_  `' F )  ->  ( `' F  |`  dom  `' ( F  |`  A ) )  =  `' ( F  |`  A )
)
97, 8mpan2 421 . 2  |-  ( Fun  `' F  ->  ( `' F  |`  dom  `' ( F  |`  A )
)  =  `' ( F  |`  A )
)
104, 9syl5req 2163 1  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    C_ wss 3041   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511   "cima 4512   Fun wfun 5087
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095
This theorem is referenced by:  cnvresid  5167  funcnvres2  5168  f1orescnv  5351  f1imacnv  5352  sbthlemi4  6816  hmeores  12411
  Copyright terms: Public domain W3C validator