Theorem relinxp 5815

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 5695	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5814	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3948   × cxp 5675  Rel wrel 5682

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by:  elinxp  6020  cnvcnv  6192  tpostpos  8231  erinxp  8785  brdom3  10523  brdom5  10524  brdom4  10525  fpwwe2lem7  10632  fpwwe2lem8  10633  fpwwe2lem11  10636  pwsle  17438  opsrtoslem2  21617  elrn3  34732  bj-idres  36041  br1cnvinxp  37124  inxprnres  37161  inxpss  37180  inxpss2  37184  iss2  37213  inxp2  37236  inxpxrn  37265