Theorem relin1 5786

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 4181	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5754	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∩ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3897   ⊆ wss 3898  Rel wrel 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3905  df-ss 3915  df-rel 5654

This theorem is used by:  inopab  5803  idsset  36574  dihmeetlem1N  42267  dihglblem5apreN  42268  dihmeetlem4preN  42283  dihmeetlem13N  42296  uptrlem2  50241  uptra  50245  uptrar  50246  uptr2a  50252  thincciso2  50485