Theorem dmdju 32631

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 5857	. . 3 ⊢ dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵)
2		dmdju.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
3		dmxp 5873	. . . . 5 ⊢ (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
4	2, 3	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
5	4	iuneq2dv 4966	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
6	1, 5	eqtrid 2780	. 2 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
7		iunid 5011	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
8	6, 7	eqtrdi 2784	1 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∅c0 4282  {csn 4575  ∪ ciun 4941   × cxp 5617  dom cdm 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-iun 4943  df-br 5094  df-opab 5156  df-xp 5625  df-dm 5629

This theorem is referenced by:  gsumwrd2dccat  33054