Theorem funin 5408

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 3429	. 2 |- ( F i^i G ) C_ F
2		funss 5352	. 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 3200   C_ wss 3201   Fun wfun 5327

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-fun 5335

This theorem is referenced by: (None)