Theorem iuneq1i

Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021) Remove DV conditions. (Revised by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	iuneq1i.1	⊢ 𝐴 = 𝐵
Assertion	iuneq1i	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶

Proof

Step	Hyp	Ref	Expression
1		iuneq1i.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	anbi1i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶 ) )
4	3	rexbii2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 )
5	4	abbii	⊢ { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 } = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
6		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 }
7		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
8	5 6 7	3eqtr4i	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶