Theorem ptpjpre2

Description: The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015)

	Ref	Expression
Hypotheses	ptbas.1	⊢ 𝐵 = { 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 ) ) }
	ptbasfi.2	⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
Assertion	ptpjpre2	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝐼 ) ) “ 𝑈 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ptbas.1	⊢ 𝐵 = { 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 ) ) }
2		ptbasfi.2	⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 )
3	2	ptpjpre1	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝐼 ) ) “ 𝑈 ) = X 𝑛 ∈ 𝐴 if ( 𝑛 = 𝐼 , 𝑈 , ∪ ( 𝐹 ‘ 𝑛 ) ) )
4		simpll	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → 𝐴 ∈ 𝑉 )
5		snfi	⊢ { 𝐼 } ∈ Fin
6	5	a1i	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → { 𝐼 } ∈ Fin )
7		simprr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) )
8	7	ad2antrr	⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) ∧ 𝑛 = 𝐼 ) → 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) )
9		simpr	⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) ∧ 𝑛 = 𝐼 ) → 𝑛 = 𝐼 )
10	9	fveq2d	⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) ∧ 𝑛 = 𝐼 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝐼 ) )
11	8 10	eleqtrrd	⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) ∧ 𝑛 = 𝐼 ) → 𝑈 ∈ ( 𝐹 ‘ 𝑛 ) )
12		simplr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → 𝐹 : 𝐴 ⟶ Top )
13	12	ffvelcdmda	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑛 ) ∈ Top )
14		eqid	⊢ ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑛 )
15	14	topopn	⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ Top → ∪ ( 𝐹 ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) )
16	13 15	syl	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → ∪ ( 𝐹 ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) )
17	16	adantr	⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) ∧ ¬ 𝑛 = 𝐼 ) → ∪ ( 𝐹 ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) )
18	11 17	ifclda	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → if ( 𝑛 = 𝐼 , 𝑈 , ∪ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 𝐹 ‘ 𝑛 ) )
19		eldifsni	⊢ ( 𝑛 ∈ ( 𝐴 ∖ { 𝐼 } ) → 𝑛 ≠ 𝐼 )
20	19	neneqd	⊢ ( 𝑛 ∈ ( 𝐴 ∖ { 𝐼 } ) → ¬ 𝑛 = 𝐼 )
21	20	adantl	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∖ { 𝐼 } ) ) → ¬ 𝑛 = 𝐼 )
22	21	iffalsed	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∖ { 𝐼 } ) ) → if ( 𝑛 = 𝐼 , 𝑈 , ∪ ( 𝐹 ‘ 𝑛 ) ) = ∪ ( 𝐹 ‘ 𝑛 ) )
23	1 4 6 18 22	elptr2	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → X 𝑛 ∈ 𝐴 if ( 𝑛 = 𝐼 , 𝑈 , ∪ ( 𝐹 ‘ 𝑛 ) ) ∈ 𝐵 )
24	3 23	eqeltrd	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑈 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝐼 ) ) “ 𝑈 ) ∈ 𝐵 )