Theorem dmdju

Description: Domain of a disjoint union of non-empty sets. (Contributed by Thierry Arnoux, 5-Oct-2025)

	Ref	Expression
Hypothesis	dmdju.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≠ ∅ )
Assertion	dmdju	⊢ ( 𝜑 → dom ∪ 𝑥 ∈ 𝐴 ( { 𝑥 } × 𝐵 ) = 𝐴 )

Proof

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