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Theorem fnresdm 5036
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 5025 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 5026 . . 3 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 eqimss 3025 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
42, 3syl 14 . 2 (𝐹 Fn 𝐴 → dom 𝐹𝐴)
5 relssres 4676 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐴) → (𝐹𝐴) = 𝐹)
61, 4, 5syl2anc 397 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2945  dom cdm 4373  cres 4375  Rel wrel 4378   Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-dm 4383  df-res 4385  df-fun 4932  df-fn 4933
This theorem is referenced by:  fnima  5045  fresin  5096  resasplitss  5097  fsnunfv  5391  fsnunres  5392  fseq1p1m1  9058  facnn  9595  fac0  9596
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