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Metamath Proof Explorer

Theorem fgss

Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion fgss ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑋 filGen 𝐹 ) ⊆ ( 𝑋 filGen 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ssrexv ⊢ ( 𝐹 ⊆ 𝐺 → ( ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → ∃ 𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡 ) )
2 1 anim2d ⊢ ( 𝐹 ⊆ 𝐺 → ( ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) → ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡 ) ) )
3 2 3ad2ant3 ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) → ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡 ) ) )
4 elfg ⊢ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
5 4 3ad2ant1 ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ) ) )
6 elfg ⊢ ( 𝐺 ∈ ( fBas ‘ 𝑋 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐺 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡 ) ) )
7 6 3ad2ant2 ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐺 ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑥 ∈ 𝐺 𝑥 ⊆ 𝑡 ) ) )
8 3 5 7 3imtr4d ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑡 ∈ ( 𝑋 filGen 𝐹 ) → 𝑡 ∈ ( 𝑋 filGen 𝐺 ) ) )
9 8 ssrdv ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐺 ∈ ( fBas ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑋 filGen 𝐹 ) ⊆ ( 𝑋 filGen 𝐺 ) )