Theorem relin1 5803

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 4197	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5772	. 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 3912   ⊆ wss 3913  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-in 3920  df-ss 3930  df-rel 5672

This theorem is referenced by:  inopab  5820  idsset  36338  dihmeetlem1N  42014  dihglblem5apreN  42015  dihmeetlem4preN  42030  dihmeetlem13N  42043  uptrlem2  49938  uptra  49942  uptrar  49943  uptr2a  49949  thincciso2  50182