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Theorem fnresi 5543
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 5463 . . 3  |-  Fun  _I
2 funres 5472 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 5 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5176 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5436 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 911 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    _I cid 4646   dom cdm 4855    |` cres 4857   Fun wfun 5427    Fn wfn 5428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-br 4308  df-opab 4366  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-res 4867  df-fun 5435  df-fn 5436
This theorem is referenced by:  f1oi  5691  fninfp  5920  fndifnfp  5922  fnnfpeq0  5924  fveqf1o  6015  weniso  6060  iordsmo  6833  fipreima  7632  dfac9  8320  pmtrfinv  15982  ustuqtop3  19833  fta1blem  21655  qaa  21804  dfiop2  25172  cvmliftlem4  27192  cvmliftlem5  27193  dvsid  29624  dflinc2  30963  ltrnid  33798
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