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Theorem fnresi 5502
Description: Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
fnresi  |-  (  _I  |`  A )  Fn  A

Proof of Theorem fnresi
StepHypRef Expression
1 funi 5423 . . 3  |-  Fun  _I
2 funres 5432 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 8 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5136 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5397 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 887 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    _I cid 4434   dom cdm 4818    |` cres 4820   Fun wfun 5388    Fn wfn 5389
This theorem is referenced by:  f1oi  5653  fveqf1o  5968  weniso  6014  iordsmo  6555  fipreima  7347  dfac9  7949  ustuqtop3  18194  fta1blem  19958  qaa  20107  dfiop2  23104  cvmliftlem4  24754  cvmliftlem5  24755  fninfp  26426  fndifnfp  26428  fnnfpeq0  26430  pmtrfinv  27071  dvsid  27217  ltrnid  30249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-fun 5396  df-fn 5397
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