Theorem fnresdm 5467

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 5454	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 5455	. . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		eqimss 3292	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴)
5		relssres 5076	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹)
6	1, 4, 5	syl2anc 411	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   = wceq 1398   ⊆ wss 3211  dom cdm 4749   ↾ cres 4751  Rel wrel 4754   Fn wfn 5347

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-dm 4759  df-res 4761  df-fun 5354  df-fn 5355

This theorem is referenced by:  fnima  5477  fresin  5543  resasplitss  5544  fresaunres2disj  5545  fnsnsplitss  5883  fsnunfv  5885  fsnunres  5886  fnsnsplitdc  6738  mapunen  7104  fnfi  7203  fseq1p1m1  10428  facnn  11089  fac0  11090  rnrhmsubrg  14397  cnfldms  15401  dfrelog  15725  eupthvdres  16470  domomsubct  16775  gfsump1  16868  gfsumcl  16870