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Theorem fnresdm 4934
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 4923 . 2 (𝐹 Fn A → Rel 𝐹)
2 fndm 4924 . . 3 (𝐹 Fn A → dom 𝐹 = A)
3 eqimss 2974 . . 3 (dom 𝐹 = A → dom 𝐹A)
42, 3syl 14 . 2 (𝐹 Fn A → dom 𝐹A)
5 relssres 4575 . 2 ((Rel 𝐹 dom 𝐹A) → (𝐹A) = 𝐹)
61, 4, 5syl2anc 393 1 (𝐹 Fn A → (𝐹A) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wss 2894  dom cdm 4272  cres 4274  Rel wrel 4277   Fn wfn 4824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-dm 4282  df-res 4284  df-fun 4831  df-fn 4832
This theorem is referenced by:  fnima  4943  fresin  4993  resasplitss  4994  fsnunfv  5288  fsnunres  5289
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