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Theorem dmdju 32657
Description: Domain of a disjoint union of non-empty sets. (Contributed by Thierry Arnoux, 5-Oct-2025.)
Hypothesis
Ref Expression
dmdju.1 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
Assertion
Ref Expression
dmdju (𝜑 → dom 𝑥𝐴 ({𝑥} × 𝐵) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem dmdju
StepHypRef Expression
1 dmiun 5924 . . 3 dom 𝑥𝐴 ({𝑥} × 𝐵) = 𝑥𝐴 dom ({𝑥} × 𝐵)
2 dmdju.1 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
3 dmxp 5939 . . . . 5 (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
42, 3syl 17 . . . 4 ((𝜑𝑥𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
54iuneq2dv 5016 . . 3 (𝜑 𝑥𝐴 dom ({𝑥} × 𝐵) = 𝑥𝐴 {𝑥})
61, 5eqtrid 2789 . 2 (𝜑 → dom 𝑥𝐴 ({𝑥} × 𝐵) = 𝑥𝐴 {𝑥})
7 iunid 5060 . 2 𝑥𝐴 {𝑥} = 𝐴
86, 7eqtrdi 2793 1 (𝜑 → dom 𝑥𝐴 ({𝑥} × 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  {csn 4626   ciun 4991   × cxp 5683  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695
This theorem is referenced by:  gsumwrd2dccat  33070
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