Theorem funin 6494

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 ∩ 𝐺))

Proof of Theorem funin

Step	Hyp	Ref	Expression
1		inss1 4159	. 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		funss 6437	. 2 ⊢ ((𝐹 ∩ 𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)))
3	1, 2	ax-mp 5	1 ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))

Syntax hints:   → wi 4   ∩ cin 3882   ⊆ wss 3883  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-fun 6420

This theorem is referenced by: (None)