Theorem funin 6424

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 4204	. 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		funss 6368	. 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 3934   ⊆ wss 3935  Fun wfun 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-ss 3951  df-br 5059  df-opab 5121  df-rel 5556  df-cnv 5557  df-co 5558  df-fun 6351

This theorem is referenced by: (None)