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Metamath Proof Explorer

Theorem ptopn

Description: A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015)

Ref Expression
Hypotheses ptopn.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
ptopn.2 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Top )
ptopn.3 ⊢ ( 𝜑 → 𝑊 ∈ Fin )
ptopn.4 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑆 ∈ ( 𝐹 ‘ 𝑘 ) )
ptopn.5 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑊 ) ) → 𝑆 = ∪ ( 𝐹 ‘ 𝑘 ) )
Assertion ptopn ( 𝜑 → X 𝑘 ∈ 𝐴 𝑆 ∈ ( ∏t ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 ptopn.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2 ptopn.2 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Top )
3 ptopn.3 ⊢ ( 𝜑 → 𝑊 ∈ Fin )
4 ptopn.4 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑆 ∈ ( 𝐹 ‘ 𝑘 ) )
5 ptopn.5 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑊 ) ) → 𝑆 = ∪ ( 𝐹 ‘ 𝑘 ) )
6 eqid ⊢ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) }
7 6 ptbas ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases )
8 1 2 7 syl2anc ⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases )
9 bastg ⊢ ( { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) )
10 8 9 syl ⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) )
11 2 ffnd ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
12 6 ptval ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ) → ( ∏t ‘ 𝐹 ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) )
13 1 11 12 syl2anc ⊢ ( 𝜑 → ( ∏t ‘ 𝐹 ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) )
14 10 13 sseqtrrd ⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ ( ∏t ‘ 𝐹 ) )
15 6 1 3 4 5 elptr2 ⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝑆 ∈ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } )
16 14 15 sseldd ⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝑆 ∈ ( ∏t ‘ 𝐹 ) )