Theorem relin1 5789

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 4133	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		relss 5757	. 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 3895   ∩ cin 4089  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-in 4091  df-rel 5657

This theorem is used by:  inopab  5806  idsset  36652  dihmeetlem1N  42348  dihglblem5apreN  42349  dihmeetlem4preN  42364  dihmeetlem13N  42377  uptrlem2  50319  uptra  50323  uptrar  50324  uptr2a  50330  thincciso2  50563