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Theorem fnresi 5288
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 5203 . . 3  |-  Fun  _I
2 funres 5212 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 5 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 4922 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5174 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 927 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff set class
Syntax hints:    = wceq 1335    _I cid 4249   dom cdm 4587    |` cres 4589   Fun wfun 5165    Fn wfn 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-res 4599  df-fun 5173  df-fn 5174
This theorem is referenced by:  f1oi  5453  iordsmo  6245  omp1eomlem  7039  ctm  7054
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