Theorem funres 5393

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 5062	. 2 |- ( F |` A ) C_ F
2		funss 5371	. 2 |- ( ( F |` A ) C_ F -> ( Fun F -> Fun ( F |` A ) ) )
3	1, 2	ax-mp 5	1 |- ( Fun F -> Fun ( F |` A ) )

Syntax hints:   -> wi 4   C_ wss 3211   |` cres 4751   Fun wfun 5346

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-res 4761  df-fun 5354

This theorem is referenced by:  funresd  5394  fnssresb  5470  fnresi  5476  fores  5600  respreima  5805  resfunexg  5905  funfvima  5918  smores  6523  smores2  6525  frecfun  6626  residfi  7207  sbthlem7  7233  setsfun  13247  setsfun0  13248  uhgrspansubgrlem  16271