Theorem istop2g

Description: Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg . (Contributed by NM, 19-Jul-2006)

	Ref	Expression
Assertion	istop2g	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin ) → ∩ 𝑥 ∈ 𝐽 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		istopg	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ) )
2		fiint	⊢ ( ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin ) → ∩ 𝑥 ∈ 𝐽 ) )
3	2	anbi2i	⊢ ( ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∩ 𝑦 ) ∈ 𝐽 ) ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin ) → ∩ 𝑥 ∈ 𝐽 ) ) )
4	1 3	syl6bb	⊢ ( 𝐽 ∈ 𝐴 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽 ) ∧ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin ) → ∩ 𝑥 ∈ 𝐽 ) ) ) )