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Theorem dmxpm 4759
 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 2202 . . 3 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
21cbvexv 1890 . 2 (∃𝑥 𝑥𝐵 ↔ ∃𝑧 𝑧𝐵)
3 df-xp 4545 . . . 4 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
43dmeqi 4740 . . 3 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
5 id 19 . . . . 5 (∃𝑧 𝑧𝐵 → ∃𝑧 𝑧𝐵)
65ralrimivw 2506 . . . 4 (∃𝑧 𝑧𝐵 → ∀𝑦𝐴𝑧 𝑧𝐵)
7 dmopab3 4752 . . . 4 (∀𝑦𝐴𝑧 𝑧𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
86, 7sylib 121 . . 3 (∃𝑧 𝑧𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
94, 8syl5eq 2184 . 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 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  {copab 3988   × cxp 4537  dom cdm 4539 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-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-dm 4549 This theorem is referenced by:  rnxpm  4968  ssxpbm  4974  ssxp1  4975  xpexr2m  4980  relrelss  5065  unixpm  5074  exmidfodomrlemim  7062
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