Theorem relinxp 5724

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 5607	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5723	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3886   × cxp 5587  Rel wrel 5594

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596

This theorem is referenced by:  elinxp  5929  cnvcnv  6095  tpostpos  8062  erinxp  8580  brdom3  10284  brdom5  10285  brdom4  10286  fpwwe2lem7  10393  fpwwe2lem8  10394  fpwwe2lem11  10397  pwsle  17203  opsrtoslem2  21263  elrn3  33729  bj-idres  35331  inxprnres  36427  inxpss  36447  inxpss2  36450  iss2  36479  inxp2  36497  inxpxrn  36521