Theorem funres 4961

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 4653	. 2 |- ( F |` A ) C_ F
2		funss 4940	. 2 |- ( ( F |` A ) C_ F -> ( Fun F -> Fun ( F |` A ) ) )
3	1, 2	ax-mp 7	1 |- ( Fun F -> Fun ( F |` A ) )

Syntax hints:   -> wi 4   C_ wss 2973   |` cres 4365   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-rel 4370  df-cnv 4371  df-co 4372  df-res 4375  df-fun 4924

This theorem is referenced by:  fnssresb  5031  fnresi  5036  fores  5135  respreima  5316  resfunexg  5403  funfvima  5411  smores  5930  smores2  5932