Theorem funin 6621

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 4132	. 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		funss 6563	. 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   ⊆ wss 3896   ∩ cin 4088  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-in 4090  df-br 5103  df-opab 5167  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6545

This theorem is used by: (None)