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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)

Assertion

Ref	Expression
dmdju	⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem dmdju

Step	Hyp	Ref	Expression
1		dmiun 5924	. . 3 ⊢ dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵)
2		dmdju.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
3		dmxp 5939	. . . . 5 ⊢ (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
4	2, 3	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
5	4	iuneq2dv 5016	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
6	1, 5	eqtrid 2789	. 2 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
7		iunid 5060	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
8	6, 7	eqtrdi 2793	1 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

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