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Theorem fnresi 5313
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 5228 . . 3 Fun I
2 funres 5237 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 4944 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 5199 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 937 1 ( I ↾ 𝐴) Fn 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1348   I cid 4271  dom cdm 4609  cres 4611  Fun wfun 5190   Fn wfn 5191
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-res 4621  df-fun 5198  df-fn 5199
This theorem is referenced by:  f1oi  5478  iordsmo  6273  omp1eomlem  7067  ctm  7082
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