Theorem relinxp 5785

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 5663	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5784	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3903   × cxp 5643  Rel wrel 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-in 3911  df-ss 3921  df-opab 5162  df-xp 5651  df-rel 5652

This theorem is referenced by:  inxp  5802  elinxp  6003  cnvcnv  6174  tpostpos  8221  brinxper  8703  erinxp  8768  brdom3  10482  brdom5  10483  brdom4  10484  fpwwe2lem7  10592  fpwwe2lem8  10593  fpwwe2lem11  10596  pwsle  17505  opsrtoslem2  22089  elrn3  36076  bj-idres  37616  br1cnvinxp  38722  inxprnres  38761  inxpss  38780  inxpss2  38784  iss2  38807  inxp2  38838  inxpxrn  38881