Theorem relin1 5795

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 4136	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5763	. 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   ⊆ wss 3900   ∩ cin 4092  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-rel 5662

This theorem is used by:  inopab  5812  idsset  36829  dihmeetlem1N  42538  dihglblem5apreN  42539  dihmeetlem4preN  42554  dihmeetlem13N  42567  uptrlem2  50530  uptra  50534  uptrar  50535  uptr2a  50541  thincciso2  50793