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Theorem rnxpm 5070
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty 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 4649 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5059 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4840 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2208 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxpm 4859 . 2 (∃𝑥 𝑥𝐴 → dom (𝐵 × 𝐴) = 𝐵)
64, 5eqtrid 2232 1 (∃𝑥 𝑥𝐴 → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wex 1502  wcel 2158   × cxp 4636  ccnv 4637  dom cdm 4638  ran crn 4639
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  ssxpbm  5076  ssxp2  5078  xpexr2m  5082  xpima2m  5088  unixpm  5176  djuexb  7057  exmidfodomrlemim  7214
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