Theorem funin 4995

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 3187	. 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		funss 4944	. 2 ⊢ ((𝐹 ∩ 𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)))
3	1, 2	ax-mp 7	1 ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))

Syntax hints:   → wi 4   ∩ cin 2973   ⊆ wss 2974  Fun wfun 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-br 3788  df-opab 3842  df-rel 4372  df-cnv 4373  df-co 4374  df-fun 4928

This theorem is referenced by: (None)