Theorem iinxsng

Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012) (Proof shortened by Mario Carneiro, 17-Nov-2016)

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

Proof

Step	Hyp	Ref	Expression
1		iinxsng.1	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
2		df-iin	⊢ ∩ 𝑥 ∈ { 𝐴 } 𝐵 = { 𝑦 ∣ ∀ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 }
3	1	eleq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
4	3	ralsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
5	4	abbi1dv	⊢ ( 𝐴 ∈ 𝑉 → { 𝑦 ∣ ∀ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 } = 𝐶 )
6	2 5	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 )