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Metamath Proof Explorer

Theorem neival

Description: Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis neifval.1 ⊢ 𝑋 = ∪ 𝐽
Assertion neival ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )

Proof

Step Hyp Ref Expression
1 neifval.1 ⊢ 𝑋 = ∪ 𝐽
2 1 neifval ⊢ ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )
3 2 fveq1d ⊢ ( 𝐽 ∈ Top → ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) = ( ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ‘ 𝑆 ) )
4 3 adantr ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) = ( ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ‘ 𝑆 ) )
5 eqid ⊢ ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )
6 cleq1lem ⊢ ( 𝑥 = 𝑆 → ( ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ↔ ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ) )
7 6 rexbidv ⊢ ( 𝑥 = 𝑆 → ( ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ↔ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ) )
8 7 rabbidv ⊢ ( 𝑥 = 𝑆 → { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )
9 1 topopn ⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 )
10 elpw2g ⊢ ( 𝑋 ∈ 𝐽 → ( 𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋 ) )
11 9 10 syl ⊢ ( 𝐽 ∈ Top → ( 𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋 ) )
12 11 biimpar ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ∈ 𝒫 𝑋 )
13 pwexg ⊢ ( 𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V )
14 rabexg ⊢ ( 𝒫 𝑋 ∈ V → { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ∈ V )
15 9 13 14 3syl ⊢ ( 𝐽 ∈ Top → { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ∈ V )
16 15 adantr ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ∈ V )
17 5 8 12 16 fvmptd3 ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ‘ 𝑆 ) = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )
18 4 17 eqtrd ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )