Theorem funres 5295

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 4966	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		funss 5273	. 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 3153   ↾ cres 4661  Fun wfun 5248

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-res 4671  df-fun 5256

This theorem is referenced by:  fnssresb  5366  fnresi  5371  fores  5486  respreima  5686  resfunexg  5779  funfvima  5790  smores  6345  smores2  6347  frecfun  6448  residfi  6999  sbthlem7  7022  setsfun  12653  setsfun0  12654