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Theorem fnresdm 5138
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 5127 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 5128 . . 3 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 eqimss 3081 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
42, 3syl 14 . 2 (𝐹 Fn 𝐴 → dom 𝐹𝐴)
5 relssres 4765 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐴) → (𝐹𝐴) = 𝐹)
61, 4, 5syl2anc 404 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  wss 3002  dom cdm 4454  cres 4456  Rel wrel 4459   Fn wfn 5025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-dm 4464  df-res 4466  df-fun 5032  df-fn 5033
This theorem is referenced by:  fnima  5147  fresin  5204  resasplitss  5205  fnsnsplitss  5512  fsnunfv  5514  fsnunres  5515  fnsnsplitdc  6280  fnfi  6702  fseq1p1m1  9571  facnn  10198  fac0  10199
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