Theorem relin1 5787

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 4190	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5756	. 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 3905   ⊆ wss 3906  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-in 3913  df-ss 3923  df-rel 5656

This theorem is referenced by:  inopab  5804  idsset  36243  dihmeetlem1N  41919  dihglblem5apreN  41920  dihmeetlem4preN  41935  dihmeetlem13N  41948  uptrlem2  49837  uptra  49841  uptrar  49842  uptr2a  49848  thincciso2  50081