Theorem fnresdm 5435

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 5422	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 5423	. . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		eqimss 3278	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴)
5		relssres 5046	. 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 1395   ⊆ wss 3197  dom cdm 4720   ↾ cres 4722  Rel wrel 4725   Fn wfn 5316

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4726  df-rel 4727  df-dm 4730  df-res 4732  df-fun 5323  df-fn 5324

This theorem is referenced by:  fnima  5445  fresin  5509  resasplitss  5510  fnsnsplitss  5845  fsnunfv  5847  fsnunres  5848  fnsnsplitdc  6664  fnfi  7119  fseq1p1m1  10307  facnn  10966  fac0  10967  rnrhmsubrg  14237  cnfldms  15231  dfrelog  15555  domomsubct  16480