Theorem relinxp 5649

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 5535	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5648	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3853   × cxp 5515  Rel wrel 5522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3077  df-v 3409  df-in 3861  df-ss 3871  df-opab 5088  df-xp 5523  df-rel 5524

This theorem is referenced by:  elinxp  5854  cnvcnv  6014  tpostpos  7915  erinxp  8374  brdom3  9973  brdom5  9974  brdom4  9975  fpwwe2lem7  10082  fpwwe2lem8  10083  fpwwe2lem11  10086  pwsle  16808  opsrtoslem2  20801  elrn3  33230  bj-idres  34840  inxprnres  35974  inxpss  35994  inxpss2  35997  iss2  36026  inxp2  36044  inxpxrn  36068