Theorem dmdju 32663

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 5926	. . 3 ⊢ dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵)
2		dmdju.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
3		dmxp 5941	. . . . 5 ⊢ (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
4	2, 3	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
5	4	iuneq2dv 5020	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
6	1, 5	eqtrid 2786	. 2 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
7		iunid 5064	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
8	6, 7	eqtrdi 2790	1 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∅c0 4338  {csn 4630  ∪ ciun 4995   × cxp 5686  dom cdm 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-iun 4997  df-br 5148  df-opab 5210  df-xp 5694  df-dm 5698

This theorem is referenced by:  gsumwrd2dccat  33052