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Theorem funcnvres 5337
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 4718 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 df-rn 4716 . . . 4  |-  ran  ( F  |`  A )  =  dom  `' ( F  |`  A )
31, 2eqtri 2316 . . 3  |-  ( F
" A )  =  dom  `' ( F  |`  A )
43reseq2i 4968 . 2  |-  ( `' F  |`  ( F " A ) )  =  ( `' F  |`  dom  `' ( F  |`  A ) )
5 resss 4995 . . . 4  |-  ( F  |`  A )  C_  F
6 cnvss 4870 . . . 4  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
75, 6ax-mp 8 . . 3  |-  `' ( F  |`  A )  C_  `' F
8 funssres 5310 . . 3  |-  ( ( Fun  `' F  /\  `' ( F  |`  A )  C_  `' F )  ->  ( `' F  |`  dom  `' ( F  |`  A ) )  =  `' ( F  |`  A )
)
97, 8mpan2 652 . 2  |-  ( Fun  `' F  ->  ( `' F  |`  dom  `' ( F  |`  A )
)  =  `' ( F  |`  A )
)
104, 9syl5req 2341 1  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265
This theorem is referenced by:  cnvresid  5338  funcnvres2  5339  f1orescnv  5504  f1imacnv  5505  sbthlem4  6990  fpwwe2lem6  8273  fpwwe2lem9  8276  hmeores  17478  dvcnvrelem2  19381  dfrelog  19939  efopnlem2  20020  diophrw  26941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
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