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Theorem rnxpm 4968
 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 4550 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 4957 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4740 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2160 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxpm 4759 . 2 (∃𝑥 𝑥𝐴 → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2184 1 (∃𝑥 𝑥𝐴 → ran (𝐴 × 𝐵) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ∃wex 1468   ∈ wcel 1480   × cxp 4537  ◡ccnv 4538  dom cdm 4539  ran crn 4540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550 This theorem is referenced by:  ssxpbm  4974  ssxp2  4976  xpexr2m  4980  xpima2m  4986  unixpm  5074  djuexb  6929  exmidfodomrlemim  7062
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