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Theorem cnvresid 5155
Description: Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
Assertion
Ref Expression
cnvresid  |-  `' (  _I  |`  A )  =  (  _I  |`  A )

Proof of Theorem cnvresid
StepHypRef Expression
1 cnvi 4901 . . 3  |-  `'  _I  =  _I
21eqcomi 2119 . 2  |-  _I  =  `'  _I
3 funi 5113 . . 3  |-  Fun  _I
4 funeq 5101 . . 3  |-  (  _I  =  `'  _I  ->  ( Fun  _I  <->  Fun  `'  _I  ) )
53, 4mpbii 147 . 2  |-  (  _I  =  `'  _I  ->  Fun  `'  _I  )
6 funcnvres 5154 . . 3  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  ( `'  _I  |`  (  _I  " A ) ) )
7 imai 4853 . . . 4  |-  (  _I  " A )  =  A
81, 7reseq12i 4775 . . 3  |-  ( `'  _I  |`  (  _I  " A ) )  =  (  _I  |`  A )
96, 8syl6eq 2163 . 2  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  (  _I  |`  A ) )
102, 5, 9mp2b 8 1  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1314    _I cid 4170   `'ccnv 4498    |` 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:  fcoi1  5261  f1oi  5361  ssidcn  12221
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