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Theorem cnvresid 5088
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 4836 . . 3  |-  `'  _I  =  _I
21eqcomi 2092 . 2  |-  _I  =  `'  _I
3 funi 5046 . . 3  |-  Fun  _I
4 funeq 5035 . . 3  |-  (  _I  =  `'  _I  ->  ( Fun  _I  <->  Fun  `'  _I  ) )
53, 4mpbii 146 . 2  |-  (  _I  =  `'  _I  ->  Fun  `'  _I  )
6 funcnvres 5087 . . 3  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  ( `'  _I  |`  (  _I  " A ) ) )
7 imai 4788 . . . 4  |-  (  _I  " A )  =  A
81, 7reseq12i 4711 . . 3  |-  ( `'  _I  |`  (  _I  " A ) )  =  (  _I  |`  A )
96, 8syl6eq 2136 . 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 1289    _I cid 4115   `'ccnv 4437    |` cres 4440   "cima 4441   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  fcoi1  5191  f1oi  5291
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