Theorem funin 5164

Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 19-Mar-2004.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

Ref	Expression
funin	⊢ (Fun F → Fun (F ∩ G))

Proof of Theorem funin

Step	Hyp	Ref	Expression
1		inss1 3476	. 2 ⊢ (F ∩ G) ⊆ F
2		funss 5127	. 2 ⊢ ((F ∩ G) ⊆ F → (Fun F → Fun (F ∩ G)))
3	1, 2	ax-mp 5	1 ⊢ (Fun F → Fun (F ∩ G))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3209   ⊆ wss 3258  Fun wfun 4776

This proof depends on 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 proof 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-fun 4790

This theorem is used by: (None)