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Theorem fnresdm 5305
Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
Assertion
Ref Expression
fnresdm (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)

Proof of Theorem fnresdm
StepHypRef Expression
1 fnrel 5294 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 5295 . . 3 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 eqimss 3201 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
42, 3syl 14 . 2 (𝐹 Fn 𝐴 → dom 𝐹𝐴)
5 relssres 4927 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐴) → (𝐹𝐴) = 𝐹)
61, 4, 5syl2anc 409 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wss 3121  dom cdm 4609  cres 4611  Rel wrel 4614   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-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-xp 4615  df-rel 4616  df-dm 4619  df-res 4621  df-fun 5198  df-fn 5199
This theorem is referenced by:  fnima  5314  fresin  5374  resasplitss  5375  fnsnsplitss  5692  fsnunfv  5694  fsnunres  5695  fnsnsplitdc  6481  fnfi  6910  fseq1p1m1  10037  facnn  10648  fac0  10649  dfrelog  13496
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