Theorem funres 5172

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 4851	. 2 |- ( F |` A ) C_ F
2		funss 5150	. 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 3076   |` cres 4549   Fun wfun 5125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-br 3938  df-opab 3998  df-rel 4554  df-cnv 4555  df-co 4556  df-res 4559  df-fun 5133

This theorem is referenced by:  fnssresb  5243  fnresi  5248  fores  5362  respreima  5556  resfunexg  5649  funfvima  5657  smores  6197  smores2  6199  frecfun  6300  sbthlem7  6859  setsfun  12033  setsfun0  12034