Theorem relinxp 5807

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 5687	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5806	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3942   × cxp 5667  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-opab 5204  df-xp 5675  df-rel 5676

This theorem is referenced by:  elinxp  6012  cnvcnv  6184  tpostpos  8229  erinxp  8784  brdom3  10522  brdom5  10523  brdom4  10524  fpwwe2lem7  10631  fpwwe2lem8  10632  fpwwe2lem11  10635  pwsle  17444  opsrtoslem2  21954  elrn3  35264  bj-idres  36547  br1cnvinxp  37636  inxprnres  37673  inxpss  37692  inxpss2  37696  iss2  37725  inxp2  37748  inxpxrn  37777