Theorem iunxsng

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016)

	Ref	Expression
Hypothesis	iunxsng.1	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
Assertion	iunxsng	⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iunxsng.1	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
2		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ { 𝐴 } 𝐵 ↔ ∃ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 )
3	1	eleq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
4	3	rexsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
5	2 4	syl5bb	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 ∈ ∪ 𝑥 ∈ { 𝐴 } 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
6	5	eqrdv	⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 )