Theorem funres 5144

Description: A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 16-Aug-1994.)

Assertion

Ref	Expression
funres	|- (Fun F -> Fun (F |` A))

Proof of Theorem funres

Step	Hyp	Ref	Expression
1		resss 4989	. 2 |- (F |` A) C_ F
2		funss 5127	. 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 3258   |` cres 4775  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-res 4789  df-fun 4790

This theorem is referenced by:  fnssresb  5196  fnresi  5201  fores  5279  respreima  5411  funfvima  5460