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Theorem fnresi 5352
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 5267 . . 3 Fun I
2 funres 5276 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 4980 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 5238 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 944 1 ( I ↾ 𝐴) Fn 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1364   I cid 4306  dom cdm 4644  cres 4646  Fun wfun 5229   Fn wfn 5230
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-fun 5237  df-fn 5238
This theorem is referenced by:  f1oi  5518  iordsmo  6323  omp1eomlem  7124  ctm  7139
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