MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relinxp Structured version   Visualization version   GIF version

Theorem relinxp 5775
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
StepHypRef Expression
1 relxp 5656 . 2 Rel (𝐴 × 𝐵)
2 relin2 5774 . 2 (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
31, 2ax-mp 5 1 Rel (𝑅 ∩ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  cin 3914   × cxp 5636  Rel wrel 5643
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  elinxp  5980  cnvcnv  6149  tpostpos  8182  erinxp  8737  brdom3  10471  brdom5  10472  brdom4  10473  fpwwe2lem7  10580  fpwwe2lem8  10581  fpwwe2lem11  10584  pwsle  17381  opsrtoslem2  21479  elrn3  34374  bj-idres  35660  br1cnvinxp  36745  inxprnres  36782  inxpss  36801  inxpss2  36805  iss2  36834  inxp2  36857  inxpxrn  36886
  Copyright terms: Public domain W3C validator