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Theorem dmxpm 4613
Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmxpm (∃𝑥 𝑥𝐵 → dom (𝐴 × 𝐵) = 𝐴)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem dmxpm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . . 3 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
21cbvexv 1838 . 2 (∃𝑥 𝑥𝐵 ↔ ∃𝑧 𝑧𝐵)
3 df-xp 4406 . . . 4 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
43dmeqi 4594 . . 3 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
5 id 19 . . . . 5 (∃𝑧 𝑧𝐵 → ∃𝑧 𝑧𝐵)
65ralrimivw 2441 . . . 4 (∃𝑧 𝑧𝐵 → ∀𝑦𝐴𝑧 𝑧𝐵)
7 dmopab3 4606 . . . 4 (∀𝑦𝐴𝑧 𝑧𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
86, 7sylib 120 . . 3 (∃𝑧 𝑧𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
94, 8syl5eq 2127 . 2 (∃𝑧 𝑧𝐵 → dom (𝐴 × 𝐵) = 𝐴)
102, 9sylbi 119 1 (∃𝑥 𝑥𝐵 → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wral 2353  {copab 3864   × cxp 4398  dom cdm 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-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 4406  df-dm 4410
This theorem is referenced by:  dmxpinm  4614  xpid11m  4615  rnxpm  4813  ssxpbm  4819  ssxp1  4820  xpexr2m  4825  relrelss  4910  unixpm  4919  xpiderm  6292  exmidfodomrlemim  6729
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