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Theorem fnresi 5333
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 5248 . . 3  |-  Fun  _I
2 funres 5257 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 5 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 4962 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5219 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 942 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff set class
Syntax hints:    = wceq 1353    _I cid 4288   dom cdm 4626    |` cres 4628   Fun wfun 5210    Fn wfn 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-fun 5218  df-fn 5219
This theorem is referenced by:  f1oi  5499  iordsmo  6297  omp1eomlem  7092  ctm  7107
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