Theorem relinxp 5756

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 5638	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5755	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3897   × cxp 5618  Rel wrel 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-opab 5155  df-xp 5626  df-rel 5627

This theorem is referenced by:  elinxp  5961  cnvcnv  6130  tpostpos  8132  erinxp  8651  brdom3  10385  brdom5  10386  brdom4  10387  fpwwe2lem7  10494  fpwwe2lem8  10495  fpwwe2lem11  10498  pwsle  17300  opsrtoslem2  21369  elrn3  34018  bj-idres  35436  br1cnvinxp  36521  inxprnres  36558  inxpss  36577  inxpss2  36581  iss2  36610  inxp2  36633  inxpxrn  36662