Theorem relinxp 5441

Description: Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)

Assertion

Ref	Expression
relinxp	⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Proof of Theorem relinxp

Step	Hyp	Ref	Expression
1		relxp 5330	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5440	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3768   × cxp 5310  Rel wrel 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-opab 4906  df-xp 5318  df-rel 5319

This theorem is referenced by:  elinxp  5644  erinxp  8059  fpwwe2lem8  9747  fpwwe2lem9  9748  fpwwe2lem12  9751  pwsle  16467  inxpss  34577  inxpss2  34580  iss2  34606  inxp2  34623  inxpxrn  34647