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Metamath Proof Explorer

Theorem ptval2

Description: The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015)

Ref Expression
Hypotheses ptval2.1 ⊢ 𝐽 = ( ∏t ‘ 𝐹 )
ptval2.2 ⊢ 𝑋 = ∪ 𝐽
ptval2.3 ⊢ 𝐺 = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
Assertion ptval2 ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐽 = ( topGen ‘ ( fi ‘ ( { 𝑋 } ∪ ran 𝐺 ) ) ) )

Proof

Step Hyp Ref Expression
1 ptval2.1 ⊢ 𝐽 = ( ∏t ‘ 𝐹 )
2 ptval2.2 ⊢ 𝑋 = ∪ 𝐽
3 ptval2.3 ⊢ 𝐺 = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
4 ffn ⊢ ( 𝐹 : 𝐴 ⟶ Top → 𝐹 Fn 𝐴 )
5 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 ) ) }
6 5 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 ) ) } ) )
7 1 6 eqtrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ) → 𝐽 = ( 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 ) ) } ) )
8 4 7 sylan2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐽 = ( 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 ) ) } ) )
9 eqid ⊢ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
10 5 9 ptbasfi ⊢ ( ( 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 ) ) } = ( fi ‘ ( { X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) )
11 1 ptuni ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ 𝐽 )
12 11 2 eqtr4di ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑋 )
13 12 sneqd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) } = { 𝑋 } )
14 12 3ad2ant1 ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑋 )
15 14 mpteq1d ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) )
16 15 cnveqd ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) → ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) = ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) )
17 16 imaeq1d ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) → ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) )
18 17 mpoeq3dva ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) )
19 18 3 eqtr4di ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) = 𝐺 )
20 19 rneqd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) = ran 𝐺 )
21 13 20 uneq12d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = ( { 𝑋 } ∪ ran 𝐺 ) )
22 21 fveq2d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) = ( fi ‘ ( { 𝑋 } ∪ ran 𝐺 ) ) )
23 10 22 eqtrd ⊢ ( ( 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 ) ) } = ( fi ‘ ( { 𝑋 } ∪ ran 𝐺 ) ) )
24 23 fveq2d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 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 ) ) } ) = ( topGen ‘ ( fi ‘ ( { 𝑋 } ∪ ran 𝐺 ) ) ) )
25 8 24 eqtrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐽 = ( topGen ‘ ( fi ‘ ( { 𝑋 } ∪ ran 𝐺 ) ) ) )