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 ) ) ) )