Theorem relinxp 5812

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 5692	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5811	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3945   × cxp 5672  Rel wrel 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-in 3953  df-ss 3963  df-opab 5208  df-xp 5680  df-rel 5681

This theorem is referenced by:  inxp  5830  elinxp  6020  cnvcnv  6195  tpostpos  8253  brinxper  8755  erinxp  8812  brdom3  10562  brdom5  10563  brdom4  10564  fpwwe2lem7  10671  fpwwe2lem8  10672  fpwwe2lem11  10675  pwsle  17502  opsrtoslem2  22065  elrn3  35597  bj-idres  36880  br1cnvinxp  37967  inxprnres  38003  inxpss  38022  inxpss2  38026  iss2  38055  inxp2  38078  inxpxrn  38106  gricer  47508