Theorem isfil

Description: The predicate "is a filter." (Contributed by FL, 20-Jul-2007) (Revised by Mario Carneiro, 28-Jul-2015)

	Ref	Expression
Assertion	isfil	⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝒫 𝑋 ( ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-fil	⊢ Fil = ( 𝑧 ∈ V ↦ { 𝑓 ∈ ( fBas ‘ 𝑧 ) ∣ ∀ 𝑥 ∈ 𝒫 𝑧 ( ( 𝑓 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝑓 ) } )
2		pweq	⊢ ( 𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋 )
3	2	adantr	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑓 = 𝐹 ) → 𝒫 𝑧 = 𝒫 𝑋 )
4		ineq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ∩ 𝒫 𝑥 ) = ( 𝐹 ∩ 𝒫 𝑥 ) )
5	4	neeq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ∩ 𝒫 𝑥 ) ≠ ∅ ↔ ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ ) )
6		eleq2	⊢ ( 𝑓 = 𝐹 → ( 𝑥 ∈ 𝑓 ↔ 𝑥 ∈ 𝐹 ) )
7	5 6	imbi12d	⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝑓 ) ↔ ( ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝐹 ) ) )
8	7	adantl	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑓 = 𝐹 ) → ( ( ( 𝑓 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝑓 ) ↔ ( ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝐹 ) ) )
9	3 8	raleqbidv	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑓 = 𝐹 ) → ( ∀ 𝑥 ∈ 𝒫 𝑧 ( ( 𝑓 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝑓 ) ↔ ∀ 𝑥 ∈ 𝒫 𝑋 ( ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝐹 ) ) )
10		fveq2	⊢ ( 𝑧 = 𝑋 → ( fBas ‘ 𝑧 ) = ( fBas ‘ 𝑋 ) )
11		fvex	⊢ ( fBas ‘ 𝑧 ) ∈ V
12		elfvdm	⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → 𝑋 ∈ dom fBas )
13	1 9 10 11 12	elmptrab2	⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝒫 𝑋 ( ( 𝐹 ∩ 𝒫 𝑥 ) ≠ ∅ → 𝑥 ∈ 𝐹 ) ) )