Theorem relin1 5755

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 4188	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5725	. 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 3902   ⊆ wss 3903  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-in 3910  df-ss 3920  df-rel 5626

This theorem is referenced by:  inopab  5772  idsset  35884  dihmeetlem1N  41289  dihglblem5apreN  41290  dihmeetlem4preN  41305  dihmeetlem13N  41318  uptrlem2  49216  uptra  49220  uptrar  49221  uptr2a  49227  thincciso2  49460