Theorem funres 5421

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 5089	. 2 |- ( F |` A ) C_ F
2		funss 5399	. 2 |- ( ( F |` A ) C_ F -> ( Fun F -> Fun ( F |` A ) ) )
3	1, 2	ax-mp 5	1 |- ( Fun F -> Fun ( F |` A ) )

This proof depends on syntax axioms:   -> wi 4   C_ wss 3220   |` cres 4778   Fun wfun 5374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-br 4132  df-opab 4194  df-rel 4783  df-cnv 4784  df-co 4785  df-res 4788  df-fun 5382

This theorem is used by:  funresd  5422  fnssresb  5498  fnresi  5504  fores  5628  respreima  5839  resfunexg  5939  funfvima  5953  smores  6567  smores2  6569  frecfun  6670  residfi  7258  sbthlem7  7285  setsfun  13445  setsfun0  13446  uhgrspansubgrlem  16722