Theorem fnresdm 5472

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 5459	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 5460	. . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		eqimss 3296	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴)
5		relssres 5081	. 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 3214  dom cdm 4754   ↾ cres 4756  Rel wrel 4759   Fn wfn 5352

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-dm 4764  df-res 4766  df-fun 5359  df-fn 5360

This theorem is referenced by:  fnima  5482  fresin  5548  resasplitss  5549  fresaunres2disj  5550  fnsnsplitss  5888  fsnunfv  5890  fsnunres  5891  fnsnsplitdc  6751  mapunen  7117  fnfi  7216  fseq1p1m1  10450  facnn  11114  fac0  11115  gfsump1  14108  gfsumcl  14110  rnrhmsubrg  14498  cnfldms  15527  dfrelog  15851  eupthvdres  16596  domomsubct  16901