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Theorem cnvresid 5322
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 5085 . . 3  |-  `'  _I  =  _I
21eqcomi 2287 . 2  |-  _I  =  `'  _I
3 funi 5284 . . 3  |-  Fun  _I
4 funeq 5274 . . 3  |-  (  _I  =  `'  _I  ->  ( Fun  _I  <->  Fun  `'  _I  ) )
53, 4mpbii 202 . 2  |-  (  _I  =  `'  _I  ->  Fun  `'  _I  )
6 funcnvres 5321 . . 3  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  ( `'  _I  |`  (  _I  " A ) ) )
7 imai 5027 . . . 4  |-  (  _I  " A )  =  A
81, 7reseq12i 4953 . . 3  |-  ( `'  _I  |`  (  _I  " A ) )  =  (  _I  |`  A )
96, 8syl6eq 2331 . 2  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  (  _I  |`  A ) )
102, 5, 9mp2b 9 1  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    _I cid 4304   `'ccnv 4688    |` cres 4691   "cima 4692   Fun wfun 5249
This theorem is referenced by:  fcoi1  5415  f1oi  5511  tsrdir  14360  gicref  14735  ssidcn  16985  idqtop  17397  idhmeo  17464  relexpcnv  24029  dispos  25287  diophrw  26838  ltrncnvnid  30316  dihmeetlem1N  31480  dihglblem5apreN  31481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
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