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Theorem dmxpm 4555
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 2100 . . 3 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
21cbvexv 1795 . 2 (∃𝑥 𝑥𝐵 ↔ ∃𝑧 𝑧𝐵)
3 df-xp 4351 . . . 4 (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
43dmeqi 4536 . . 3 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)}
5 id 19 . . . . 5 (∃𝑧 𝑧𝐵 → ∃𝑧 𝑧𝐵)
65ralrimivw 2393 . . . 4 (∃𝑧 𝑧𝐵 → ∀𝑦𝐴𝑧 𝑧𝐵)
7 dmopab3 4548 . . . 4 (∀𝑦𝐴𝑧 𝑧𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
86, 7sylib 127 . . 3 (∃𝑧 𝑧𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵)} = 𝐴)
94, 8syl5eq 2084 . 2 (∃𝑧 𝑧𝐵 → dom (𝐴 × 𝐵) = 𝐴)
102, 9sylbi 114 1 (∃𝑥 𝑥𝐵 → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  wral 2306  {copab 3817   × cxp 4343  dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355
This theorem is referenced by:  dmxpinm  4556  xpid11m  4557  rnxpm  4752  ssxpbm  4756  ssxp1  4757  xpexr2m  4762  relrelss  4844  unixpm  4853  xpiderm  6177
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