Theorem relinxp 5763

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 5642	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5762	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3900   × cxp 5622  Rel wrel 5629

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-opab 5161  df-xp 5630  df-rel 5631

This theorem is referenced by:  inxp  5780  elinxp  5978  cnvcnv  6150  tpostpos  8188  brinxper  8664  erinxp  8728  brdom3  10438  brdom5  10439  brdom4  10440  fpwwe2lem7  10548  fpwwe2lem8  10549  fpwwe2lem11  10552  pwsle  17413  opsrtoslem2  22011  elrn3  35956  bj-idres  37361  br1cnvinxp  38450  inxprnres  38487  inxpss  38506  inxpss2  38510  iss2  38533  inxp2  38556  inxpxrn  38599