Theorem funres 5359

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 5029	. 2 |- ( F |` A ) C_ F
2		funss 5337	. 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 3197   |` cres 4721   Fun wfun 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-res 4731  df-fun 5320

This theorem is referenced by:  fnssresb  5435  fnresi  5441  fores  5558  respreima  5763  resfunexg  5860  funfvima  5871  smores  6438  smores2  6440  frecfun  6541  residfi  7107  sbthlem7  7130  setsfun  13067  setsfun0  13068