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 ) )