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Theorem relxp 4734
Description: A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
relxp Rel (𝐴 × 𝐵)

Proof of Theorem relxp
StepHypRef Expression
1 xpss 4733 . 2 (𝐴 × 𝐵) ⊆ (V × V)
2 df-rel 4632 . 2 (Rel (𝐴 × 𝐵) ↔ (𝐴 × 𝐵) ⊆ (V × V))
31, 2mpbir 146 1 Rel (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  Vcvv 2737  wss 3129   × cxp 4623  Rel wrel 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-opab 4064  df-xp 4631  df-rel 4632
This theorem is referenced by:  xpiindim  4763  eliunxp  4765  opeliunxp2  4766  relres  4934  restidsing  4962  codir  5016  qfto  5017  cnvcnv  5080  dfco2  5127  unixpm  5163  ressn  5168  fliftcnv  5793  fliftfun  5794  opeliunxp2f  6236  reltpos  6248  tpostpos  6262  tposfo  6269  tposf  6270  swoer  6560  xpider  6603  erinxp  6606  xpcomf1o  6822  ltrel  8015  lerel  8017  fisumcom2  11439  fprodcom2fi  11627  txuni2  13627  txdis1cn  13649  xmeter  13807  reldvg  14019
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