MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnresi Unicode version

Theorem fnresi 5363
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 5286 . . 3  |-  Fun  _I
2 funres 5295 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 8 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5007 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5260 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 886 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff set class
Syntax hints:    = wceq 1625    _I cid 4306   dom cdm 4691    |` cres 4693   Fun wfun 5251    Fn wfn 5252
This theorem is referenced by:  f1oi  5513  fveqf1o  5808  weniso  5854  iordsmo  6376  fipreima  7163  dfac9  7764  fta1blem  19556  qaa  19705  dfiop2  22335  cvmliftlem4  23821  cvmliftlem5  23822  surjsec2  25131  fninfp  26765  fndifnfp  26767  fnnfpeq0  26769  pmtrfinv  27413  dvsid  27559  ltrnid  30397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-res 4703  df-fun 5259  df-fn 5260
  Copyright terms: Public domain W3C validator