Theorem iunsn

Description: Indexed union of a singleton. Compare dfiun2 and rnmpt . (Contributed by Steven Nguyen, 7-Jun-2023)

	Ref	Expression
Assertion	iunsn	⊢ ∪ 𝑥 ∈ 𝐴 { 𝐵 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }

Proof

Step	Hyp	Ref	Expression
1		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 { 𝐵 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ { 𝐵 } }
2		velsn	⊢ ( 𝑦 ∈ { 𝐵 } ↔ 𝑦 = 𝐵 )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ { 𝐵 } ↔ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 )
4	3	abbii	⊢ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ { 𝐵 } } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }
5	1 4	eqtri	⊢ ∪ 𝑥 ∈ 𝐴 { 𝐵 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }