Theorem dmdju 33120

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 5897	. . 3 ⊢ dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵)
2		dmdju.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
3		dmxp 5913	. . . . 5 ⊢ (𝐵 ≠ ∅ → dom ({𝑥} × 𝐵) = {𝑥})
4	2, 3	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom ({𝑥} × 𝐵) = {𝑥})
5	4	iuneq2dv 4976	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 dom ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
6	1, 5	eqtrid 2807	. 2 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑥})
7		iunid 5019	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
8	6, 7	eqtrdi 2811	1 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∅c0 4279  {csn 4584  ∪ ciun 4951   × cxp 5653  dom cdm 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5661  df-dm 5665

This theorem is used by:  gsumwrd2dccat  33518