Theorem istopg

Description: Express the predicate " J is a topology". See istop2g for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006) (Revised by Mario Carneiro, 11-Nov-2013)

	Ref	Expression
Assertion	istopg	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pweq	⊢ ( 𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽 )
2		eleq2	⊢ ( 𝑧 = 𝐽 → ( ∪ 𝑥 ∈ 𝑧 ↔ ∪ 𝑥 ∈ 𝐽 ) )
3	1 2	raleqbidv	⊢ ( 𝑧 = 𝐽 → ( ∀ 𝑥 ∈ 𝒫 𝑧 ∪ 𝑥 ∈ 𝑧 ↔ ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ) )
4		eleq2	⊢ ( 𝑧 = 𝐽 → ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑧 ↔ ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) )
5	4	raleqbi1dv	⊢ ( 𝑧 = 𝐽 → ( ∀ 𝑦 ∈ 𝑧 ( 𝑥 ∩ 𝑦 ) ∈ 𝑧 ↔ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) )
6	5	raleqbi1dv	⊢ ( 𝑧 = 𝐽 → ( ∀ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑧 ( 𝑥 ∩ 𝑦 ) ∈ 𝑧 ↔ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) )
7	3 6	anbi12d	⊢ ( 𝑧 = 𝐽 → ( ( ∀ 𝑥 ∈ 𝒫 𝑧 ∪ 𝑥 ∈ 𝑧 ∧ ∀ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑧 ( 𝑥 ∩ 𝑦 ) ∈ 𝑧 ) ↔ ( ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )
8		df-top	⊢ Top = { 𝑧 ∣ ( ∀ 𝑥 ∈ 𝒫 𝑧 ∪ 𝑥 ∈ 𝑧 ∧ ∀ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑧 ( 𝑥 ∩ 𝑦 ) ∈ 𝑧 ) }
9	7 8	elab2g	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )
10		df-ral	⊢ ( ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽 ) )
11		elpw2g	⊢ ( 𝐽 ∈ 𝐴 → ( 𝑥 ∈ 𝒫 𝐽 ↔ 𝑥 ⊆ 𝐽 ) )
12	11	imbi1d	⊢ ( 𝐽 ∈ 𝐴 → ( ( 𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ↔ ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ) )
13	12	albidv	⊢ ( 𝐽 ∈ 𝐴 → ( ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ) )
14	10 13	syl5bb	⊢ ( 𝐽 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ) )
15	14	anbi1d	⊢ ( 𝐽 ∈ 𝐴 → ( ( ∀ 𝑥 ∈ 𝒫 𝐽 ∪ 𝑥 ∈ 𝐽 ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )
16	9 15	bitrd	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )