Theorem relinxp 5820

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 5700	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5819	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Syntax hints:   ∩ cin 3948   × cxp 5680  Rel wrel 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-xp 5688  df-rel 5689

This theorem is referenced by:  inxp  5838  elinxp  6028  cnvcnv  6201  tpostpos  8258  erinxp  8816  brdom3  10559  brdom5  10560  brdom4  10561  fpwwe2lem7  10668  fpwwe2lem8  10669  fpwwe2lem11  10672  pwsle  17481  opsrtoslem2  22007  elrn3  35389  bj-idres  36672  br1cnvinxp  37760  inxprnres  37796  inxpss  37815  inxpss2  37819  iss2  37848  inxp2  37871  inxpxrn  37899  gricer  47268