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Theorem funcnvres 5482
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 4848 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 df-rn 4846 . . . 4  |-  ran  ( F  |`  A )  =  dom  `' ( F  |`  A )
31, 2eqtri 2458 . . 3  |-  ( F
" A )  =  dom  `' ( F  |`  A )
43reseq2i 5102 . 2  |-  ( `' F  |`  ( F " A ) )  =  ( `' F  |`  dom  `' ( F  |`  A ) )
5 resss 5129 . . . 4  |-  ( F  |`  A )  C_  F
6 cnvss 5007 . . . 4  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
75, 6ax-mp 5 . . 3  |-  `' ( F  |`  A )  C_  `' F
8 funssres 5453 . . 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 2483 1  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    C_ wss 3323   `'ccnv 4834   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838   Fun wfun 5407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-fun 5415
This theorem is referenced by:  cnvresid  5483  funcnvres2  5484  f1orescnv  5651  f1imacnv  5652  sbthlem4  7416  fpwwe2lem6  8794  fpwwe2lem9  8797  hmeores  19319  dvcnvrelem2  21465  dfrelog  21992  efopnlem2  22077  diophrw  29050
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