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Theorem dmxpm 4729
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 2180 . . 3 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
21cbvexv 1872 . 2 (∃𝑥 𝑥𝐵 ↔ ∃𝑧 𝑧𝐵)
3 df-xp 4515 . . . 4 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
43dmeqi 4710 . . 3 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
5 id 19 . . . . 5 (∃𝑧 𝑧𝐵 → ∃𝑧 𝑧𝐵)
65ralrimivw 2483 . . . 4 (∃𝑧 𝑧𝐵 → ∀𝑦𝐴𝑧 𝑧𝐵)
7 dmopab3 4722 . . . 4 (∀𝑦𝐴𝑧 𝑧𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
86, 7sylib 121 . . 3 (∃𝑧 𝑧𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
94, 8syl5eq 2162 . 2 (∃𝑧 𝑧𝐵 → dom (𝐴 × 𝐵) = 𝐴)
102, 9sylbi 120 1 (∃𝑥 𝑥𝐵 → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  wral 2393  {copab 3958   × cxp 4507  dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-dm 4519
This theorem is referenced by:  rnxpm  4938  ssxpbm  4944  ssxp1  4945  xpexr2m  4950  relrelss  5035  unixpm  5044  exmidfodomrlemim  7025
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