Theorem ptbasin2

Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015)

	Ref	Expression
Hypothesis	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 ) ) }
Assertion	ptbasin2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ 𝐵 ) = 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	1	ptbasin	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑢 ∩ 𝑣 ) ∈ 𝐵 )
3	2	ralrimivva	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ∩ 𝑣 ) ∈ 𝐵 )
4	1	ptuni2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐵 )
5		ixpexg	⊢ ( ∀ 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∈ V → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∈ V )
6		fvex	⊢ ( 𝐹 ‘ 𝑘 ) ∈ V
7	6	uniex	⊢ ∪ ( 𝐹 ‘ 𝑘 ) ∈ V
8	7	a1i	⊢ ( 𝑘 ∈ 𝐴 → ∪ ( 𝐹 ‘ 𝑘 ) ∈ V )
9	5 8	mprg	⊢ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∈ V
10	4 9	eqeltrrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ 𝐵 ∈ V )
11		uniexb	⊢ ( 𝐵 ∈ V ↔ ∪ 𝐵 ∈ V )
12	10 11	sylibr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ∈ V )
13		inficl	⊢ ( 𝐵 ∈ V → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ∩ 𝑣 ) ∈ 𝐵 ↔ ( fi ‘ 𝐵 ) = 𝐵 ) )
14	12 13	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ∩ 𝑣 ) ∈ 𝐵 ↔ ( fi ‘ 𝐵 ) = 𝐵 ) )
15	3 14	mpbid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ 𝐵 ) = 𝐵 )