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Theorem cnvresid 5648
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 5400 . . 3  |-  `'  _I  =  _I
21eqcomi 2456 . 2  |-  _I  =  `'  _I
3 funi 5608 . . 3  |-  Fun  _I
4 funeq 5597 . . 3  |-  (  _I  =  `'  _I  ->  ( Fun  _I  <->  Fun  `'  _I  ) )
53, 4mpbii 211 . 2  |-  (  _I  =  `'  _I  ->  Fun  `'  _I  )
6 funcnvres 5647 . . 3  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  ( `'  _I  |`  (  _I  " A ) ) )
7 imai 5339 . . . 4  |-  (  _I  " A )  =  A
81, 7reseq12i 5261 . . 3  |-  ( `'  _I  |`  (  _I  " A ) )  =  (  _I  |`  A )
96, 8syl6eq 2500 . 2  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  (  _I  |`  A ) )
102, 5, 9mp2b 10 1  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    _I cid 4780   `'ccnv 4988    |` cres 4991   "cima 4992   Fun wfun 5572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-fun 5580
This theorem is referenced by:  fcoi1  5749  f1oi  5841  tsrdir  15742  gicref  16193  ssidcn  19629  idqtop  20080  idhmeo  20147  relexpcnv  28929  diophrw  30667  ltrncnvnid  35591  dihmeetlem1N  36757  dihglblem5apreN  36758
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