Theorem funres 5374

Description: A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
funres	⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))

Proof of Theorem funres

Step	Hyp	Ref	Expression
1		resss 5043	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		funss 5352	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3201   ↾ cres 4733  Fun wfun 5327

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-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-res 4743  df-fun 5335

This theorem is referenced by:  funresd  5375  fnssresb  5451  fnresi  5457  fores  5578  respreima  5783  resfunexg  5883  funfvima  5896  smores  6501  smores2  6503  frecfun  6604  residfi  7182  sbthlem7  7205  setsfun  13178  setsfun0  13179  uhgrspansubgrlem  16197