Theorem relin1 5758

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 4168	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5728	. 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 3884   ⊆ wss 3885  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3892  df-ss 3902  df-rel 5628

This theorem is referenced by:  inopab  5775  idsset  36131  dihmeetlem1N  41797  dihglblem5apreN  41798  dihmeetlem4preN  41813  dihmeetlem13N  41826  uptrlem2  49715  uptra  49719  uptrar  49720  uptr2a  49726  thincciso2  49959