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Theorem rnxpm 4812
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
rnxpm (∃𝑥 𝑥𝐴 → ran (𝐴 × 𝐵) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rnxpm
StepHypRef Expression
1 df-rn 4410 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 4802 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4593 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2103 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxpm 4612 . 2 (∃𝑥 𝑥𝐴 → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2127 1 (∃𝑥 𝑥𝐴 → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wex 1422  wcel 1434   × cxp 4397  ccnv 4398  dom cdm 4399  ran crn 4400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-rel 4406  df-cnv 4407  df-dm 4409  df-rn 4410
This theorem is referenced by:  ssxpbm  4818  ssxp2  4820  xpexr2m  4824  xpima2m  4830  unixpm  4918  exmidfodomrlemim  6728
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