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Theorem relinxp 5674
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 5560 . 2 Rel (𝐴 × 𝐵)
2 relin2 5673 . 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 3918   × cxp 5540  Rel wrel 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-opab 5115  df-xp 5548  df-rel 5549
This theorem is referenced by:  elinxp  5877  cnvcnv  6036  tpostpos  7902  erinxp  8361  brdom3  9942  brdom5  9943  brdom4  9944  fpwwe2lem8  10051  fpwwe2lem9  10052  fpwwe2lem12  10055  pwsle  16761  opsrtoslem2  20258  elrn3  33023  bj-idres  34488  inxprnres  35619  inxpss  35639  inxpss2  35642  iss2  35671  inxp2  35689  inxpxrn  35713
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