Theorem relin 2491

Description: The intersection with a relation is a relation.

Assertion

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

Proof of Theorem relin

Step	Hyp	Ref	Expression
1		inss1 1657	. 2 ⊢ (A ∩ B) ⊆ A
2		ssrel 2479	. 2 ⊢ ((A ∩ B) ⊆ A → (Rel A → Rel (A ∩ B)))
3	1, 2	ax-mp 6	1 ⊢ (Rel A → Rel (A ∩ B))

Syntax hints:   → wi 2   ∩ cin 1486   ⊆ wss 1487  Rel wrel 2415

This theorem is referenced by:  inopab 2495  inxp 2496  cnvin 2643  funin 2708  sbthcl 3361  mapdom2lem 3388  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-rel 2425

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