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Theorem funcnvres 5154
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 4512 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
2 df-rn 4510 . . . 4  |-  ran  ( F  |`  A )  =  dom  `' ( F  |`  A )
31, 2eqtri 2135 . . 3  |-  ( F
" A )  =  dom  `' ( F  |`  A )
43reseq2i 4774 . 2  |-  ( `' F  |`  ( F " A ) )  =  ( `' F  |`  dom  `' ( F  |`  A ) )
5 resss 4801 . . . 4  |-  ( F  |`  A )  C_  F
6 cnvss 4672 . . . 4  |-  ( ( F  |`  A )  C_  F  ->  `' ( F  |`  A )  C_  `' F )
75, 6ax-mp 7 . . 3  |-  `' ( F  |`  A )  C_  `' F
8 funssres 5123 . . 3  |-  ( ( Fun  `' F  /\  `' ( F  |`  A )  C_  `' F )  ->  ( `' F  |`  dom  `' ( F  |`  A ) )  =  `' ( F  |`  A )
)
97, 8mpan2 419 . 2  |-  ( Fun  `' F  ->  ( `' F  |`  dom  `' ( F  |`  A )
)  =  `' ( F  |`  A )
)
104, 9syl5req 2160 1  |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    C_ wss 3037   `'ccnv 4498   dom cdm 4499   ran crn 4500    |` cres 4501   "cima 4502   Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-fun 5083
This theorem is referenced by:  cnvresid  5155  funcnvres2  5156  f1orescnv  5339  f1imacnv  5340  sbthlemi4  6800
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