Theorem fnresdm 5487

Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Assertion

Ref	Expression
fnresdm	⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)

Proof of Theorem fnresdm

Step	Hyp	Ref	Expression
1		fnrel 5474	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 5475	. . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		eqimss 3302	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴)
5		relssres 5096	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹)
6	1, 4, 5	syl2anc 415	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   = wceq 1402   ⊆ wss 3220  dom cdm 4769   ↾ cres 4771  Rel wrel 4774   Fn wfn 5367

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779  df-res 4781  df-fun 5374  df-fn 5375

This theorem is referenced by:  fnima  5497  fresin  5563  resasplitss  5564  fresaunres2disj  5565  fnsnsplitss  5905  fsnunfv  5907  fsnunres  5908  fnsnsplitdc  6768  mapunen  7141  fnfi  7240  fseq1p1m1  10479  facnn  11143  fac0  11144  gsump1  14134  gsumclfi  14136  gsumsubmclfi  14140  rnrhmsubrg  14533  cnfldms  15560  dfrelog  15884  eupthvdres  16630  domomsubct  16945