Theorem funres 5055

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 4737	. 2 |- ( F |` A ) C_ F
2		funss 5034	. 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 2999   |` cres 4440   Fun wfun 5009

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-br 3846  df-opab 3900  df-rel 4445  df-cnv 4446  df-co 4447  df-res 4450  df-fun 5017

This theorem is referenced by:  fnssresb  5126  fnresi  5131  fores  5242  respreima  5427  resfunexg  5518  funfvima  5526  smores  6057  smores2  6059  frecfun  6160  sbthlem7  6672  setsfun  11529  setsfun0  11530