Theorem relin1 5236

Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
relin1	⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))

Proof of Theorem relin1

Step	Hyp	Ref	Expression
1		inss1 3833	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5206	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∩ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∩ cin 3573   ⊆ wss 3574  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-rel 5121

This theorem is referenced by:  inopab  5252  idsset  31997  dihmeetlem1N  36579  dihglblem5apreN  36580  dihmeetlem4preN  36595  dihmeetlem13N  36608