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Metamath Proof Explorer

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 ) )