Theorem relin1 3268

Description: The intersection with a relation is a relation.

Assertion

Ref	Expression
relin1	⊢ (Rel A → Rel (A ∩ B))

Proof of Theorem relin1

Step	Hyp	Ref	Expression
1		inss1 2233	. 2 ⊢ (A ∩ B) ⊆ A
2		relss 3252	. 2 ⊢ ((A ∩ B) ⊆ A → (Rel A → Rel (A ∩ B)))
3	1, 2	ax-mp 7	1 ⊢ (Rel A → Rel (A ∩ B))

Syntax hints:   → wi 3   ∩ cin 2049   ⊆ wss 2050  Rel wrel 3181

This theorem is referenced by:  inopab 3274  inxp 3275  cnvin 3462  funin 3572  sbthcl 4465  mapdom2lem 4499  infxpidmlem11 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-rel 3191

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