Theorem funin 5306

Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funin	|- ( Fun F -> Fun ( F i^i G ) )

Proof of Theorem funin

Step	Hyp	Ref	Expression
1		inss1 3370	. 2 |- ( F i^i G ) C_ F
2		funss 5254	. 2 |- ( ( F i^i G ) C_ F -> ( Fun F -> Fun ( F i^i G ) ) )
3	1, 2	ax-mp 5	1 |- ( Fun F -> Fun ( F i^i G ) )

Syntax hints:   -> wi 4   i^i cin 3143   C_ wss 3144   Fun wfun 5229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-fun 5237

This theorem is referenced by: (None)