Theorem tgiun

Description: The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015)

	Ref	Expression
Assertion	tgiun	⊢ ( ( 𝐵 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ ( topGen ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dfiun3g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
2	1	adantl	⊢ ( ( 𝐵 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
3		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )
4	3	rnmptss	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ⊆ 𝐵 )
5		eltg3i	⊢ ( ( 𝐵 ∈ 𝑉 ∧ ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ⊆ 𝐵 ) → ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ ( topGen ‘ 𝐵 ) )
6	4 5	sylan2	⊢ ( ( 𝐵 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ) → ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ ( topGen ‘ 𝐵 ) )
7	2 6	eqeltrd	⊢ ( ( 𝐵 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ ( topGen ‘ 𝐵 ) )