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Metamath Proof Explorer

Theorem xkotop

Description: The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015)

Ref Expression
Assertion xkotop ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) ∈ Top )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ∪ 𝑅 = ∪ 𝑅
2 eqid ⊢ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } = { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp }
3 eqid ⊢ ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) = ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } )
4 1 2 3 xkoval ⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) = ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) )
5 fibas ⊢ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ∈ TopBases
6 tgcl ⊢ ( ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ∈ TopBases → ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) ∈ Top )
7 5 6 ax-mp ⊢ ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) ∈ Top
8 4 7 eqeltrdi ⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) ∈ Top )