Theorem funin 6600

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 4208	. 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		funss 6543	. 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 3921   ⊆ wss 3922  Fun wfun 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-in 3929  df-ss 3939  df-br 5116  df-opab 5178  df-rel 5653  df-cnv 5654  df-co 5655  df-fun 6521

This theorem is referenced by: (None)