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Metamath Proof Explorer

Theorem fgfil

Description: A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion fgfil ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 filGen 𝐹 ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 filfbas ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )
2 elfg ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
3 1 2 syl ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
4 filss ⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡 ) ) → 𝑡 ∈ 𝐹 )
5 4 3exp2 ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ∈ 𝐹 → ( 𝑡 ⊆ 𝑋 → ( 𝑥 ⊆ 𝑡 → 𝑡 ∈ 𝐹 ) ) ) )
6 5 com34 ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ∈ 𝐹 → ( 𝑥 ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹 ) ) ) )
7 6 rexlimdv ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹 ) ) )
8 7 impcomd ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) → 𝑡 ∈ 𝐹 ) )
9 3 8 sylbid ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) → 𝑡 ∈ 𝐹 ) )
10 9 ssrdv ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 filGen 𝐹 ) ⊆ 𝐹 )
11 ssfg ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → 𝐹 ⊆ ( 𝑋 filGen 𝐹 ) )
12 1 11 syl ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ⊆ ( 𝑋 filGen 𝐹 ) )
13 10 12 eqssd ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 filGen 𝐹 ) = 𝐹 )