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Theorem fnresdm 5344
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 5333 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 5334 . . 3 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 eqimss 3224 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
42, 3syl 14 . 2 (𝐹 Fn 𝐴 → dom 𝐹𝐴)
5 relssres 4963 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐴) → (𝐹𝐴) = 𝐹)
61, 4, 5syl2anc 411 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wss 3144  dom cdm 4644  cres 4646  Rel wrel 4649   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-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-xp 4650  df-rel 4651  df-dm 4654  df-res 4656  df-fun 5237  df-fn 5238
This theorem is referenced by:  fnima  5353  fresin  5413  resasplitss  5414  fnsnsplitss  5735  fsnunfv  5737  fsnunres  5738  fnsnsplitdc  6529  fnfi  6965  fseq1p1m1  10123  facnn  10738  fac0  10739  dfrelog  14733
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