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Theorem relxp 4736
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 4735 . 2 (𝐴 × 𝐵) ⊆ (V × V)
2 df-rel 4634 . 2 (Rel (𝐴 × 𝐵) ↔ (𝐴 × 𝐵) ⊆ (V × V))
31, 2mpbir 146 1 Rel (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  Vcvv 2738  wss 3130   × cxp 4625  Rel wrel 4632
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 2740  df-in 3136  df-ss 3143  df-opab 4066  df-xp 4633  df-rel 4634
This theorem is referenced by:  xpiindim  4765  eliunxp  4767  opeliunxp2  4768  relres  4936  restidsing  4964  codir  5018  qfto  5019  cnvcnv  5082  dfco2  5129  unixpm  5165  ressn  5170  fliftcnv  5796  fliftfun  5797  opeliunxp2f  6239  reltpos  6251  tpostpos  6265  tposfo  6272  tposf  6273  swoer  6563  xpider  6606  erinxp  6609  xpcomf1o  6825  ltrel  8019  lerel  8021  fisumcom2  11446  fprodcom2fi  11634  txuni2  13759  txdis1cn  13781  xmeter  13939  reldvg  14151
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