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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
StepHypRef Expression
1 relxp 5700 . 2 Rel (𝐴 × 𝐵)
2 relin2 5819 . 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 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
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