Theorem relinxp 5713

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 5598	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5712	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3882   × cxp 5578  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  elinxp  5918  cnvcnv  6084  tpostpos  8033  erinxp  8538  brdom3  10215  brdom5  10216  brdom4  10217  fpwwe2lem7  10324  fpwwe2lem8  10325  fpwwe2lem11  10328  pwsle  17120  opsrtoslem2  21173  elrn3  33635  bj-idres  35258  inxprnres  36354  inxpss  36374  inxpss2  36377  iss2  36406  inxp2  36424  inxpxrn  36448