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Theorem fnresi 5067
 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 4982 . . 3 Fun I
2 funres 4991 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 7 . 2 Fun ( I ↾ 𝐴)
4 dmresi 4711 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 4955 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 884 1 ( I ↾ 𝐴) Fn 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1285   I cid 4071  dom cdm 4391   ↾ cres 4393  Fun wfun 4946   Fn wfn 4947 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-fun 4954  df-fn 4955 This theorem is referenced by:  f1oi  5215  iordsmo  5966
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