Theorem funin 5401

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 3427	. 2 |- ( F i^i G ) C_ F
2		funss 5345	. 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 3199   C_ wss 3200   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by: (None)