Theorem relinxp 5651

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 5537	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5650	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3880   × cxp 5517  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  elinxp  5856  cnvcnv  6016  tpostpos  7895  erinxp  8354  brdom3  9939  brdom5  9940  brdom4  9941  fpwwe2lem8  10048  fpwwe2lem9  10049  fpwwe2lem12  10052  pwsle  16757  opsrtoslem2  20724  elrn3  33111  bj-idres  34575  inxprnres  35709  inxpss  35729  inxpss2  35732  iss2  35761  inxp2  35779  inxpxrn  35803