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Theorem funcnvres 5655
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 5012 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 df-rn 5010 . . . 4  |-  ran  ( F  |`  A )  =  dom  `' ( F  |`  A )
31, 2eqtri 2496 . . 3  |-  ( F
" A )  =  dom  `' ( F  |`  A )
43reseq2i 5268 . 2  |-  ( `' F  |`  ( F " A ) )  =  ( `' F  |`  dom  `' ( F  |`  A ) )
5 resss 5295 . . . 4  |-  ( F  |`  A )  C_  F
6 cnvss 5173 . . . 4  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
75, 6ax-mp 5 . . 3  |-  `' ( F  |`  A )  C_  `' F
8 funssres 5626 . . 3  |-  ( ( Fun  `' F  /\  `' ( F  |`  A )  C_  `' F )  ->  ( `' F  |`  dom  `' ( F  |`  A ) )  =  `' ( F  |`  A )
)
97, 8mpan2 671 . 2  |-  ( Fun  `' F  ->  ( `' F  |`  dom  `' ( F  |`  A )
)  =  `' ( F  |`  A )
)
104, 9syl5req 2521 1  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    C_ wss 3476   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588
This theorem is referenced by:  cnvresid  5656  funcnvres2  5657  f1orescnv  5829  f1imacnv  5830  sbthlem4  7627  fpwwe2lem6  9009  fpwwe2lem9  9012  hmeores  20007  dvcnvrelem2  22154  dfrelog  22681  efopnlem2  22766  diophrw  30296
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