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Theorem fnresdm 5232
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 5221 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 5222 . . 3 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 eqimss 3151 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
42, 3syl 14 . 2 (𝐹 Fn 𝐴 → dom 𝐹𝐴)
5 relssres 4857 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐴) → (𝐹𝐴) = 𝐹)
61, 4, 5syl2anc 408 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wss 3071  dom cdm 4539  cres 4541  Rel wrel 4544   Fn wfn 5118
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549  df-res 4551  df-fun 5125  df-fn 5126
This theorem is referenced by:  fnima  5241  fresin  5301  resasplitss  5302  fnsnsplitss  5619  fsnunfv  5621  fsnunres  5622  fnsnsplitdc  6401  fnfi  6825  fseq1p1m1  9886  facnn  10485  fac0  10486  dfrelog  12963
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