Theorem funres 5416

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 5085	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		funss 5394	. 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 3220   ↾ cres 4774  Fun wfun 5369

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-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-br 4129  df-opab 4191  df-rel 4779  df-cnv 4780  df-co 4781  df-res 4784  df-fun 5377

This theorem is referenced by:  funresd  5417  fnssresb  5493  fnresi  5499  fores  5623  respreima  5830  resfunexg  5930  funfvima  5944  smores  6557  smores2  6559  frecfun  6660  residfi  7248  sbthlem7  7274  setsfun  13370  setsfun0  13371  uhgrspansubgrlem  16500