Theorem dmdju 32735

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 5862	. . 3 ⊢ dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵)
2		dmdju.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
3		dmxp 5878	. . . . 5 ⊢ (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
4	2, 3	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
5	4	iuneq2dv 4959	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
6	1, 5	eqtrid 2784	. 2 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
7		iunid 5004	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
8	6, 7	eqtrdi 2788	1 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  {csn 4568  ∪ ciun 4934   × cxp 5622  dom cdm 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-iun 4936  df-br 5087  df-opab 5149  df-xp 5630  df-dm 5634

This theorem is referenced by:  gsumwrd2dccat  33154