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Metamath Proof Explorer

Theorem ssfg

Description: A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion ssfg ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → 𝐹 ⊆ ( 𝑋 filGen 𝐹 ) )

Proof

Step Hyp Ref Expression
1 fbelss ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐹 ) → 𝑡 ⊆ 𝑋 )
2 1 ex ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋 ) )
3 ssid ⊢ 𝑡 ⊆ 𝑡
4 sseq1 ⊢ ( 𝑥 = 𝑡 → ( 𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡 ) )
5 4 rspcev ⊢ ( ( 𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡 ) → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 )
6 3 5 mpan2 ⊢ ( 𝑡 ∈ 𝐹 → ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 )
7 2 6 jca2 ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ 𝐹 → ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
8 elfg ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
9 7 8 sylibrd ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ 𝐹 → 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ) )
10 9 ssrdv ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → 𝐹 ⊆ ( 𝑋 filGen 𝐹 ) )