Theorem trfil1

Description: Conditions for the trace of a filter L to be a filter. (Contributed by FL, 2-Sep-2013) (Revised by Stefan O'Rear, 2-Aug-2015)

	Ref	Expression
Assertion	trfil1	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 = ∪ ( 𝐿 ↾t 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sseqin2	⊢ ( 𝐴 ⊆ 𝑌 ↔ ( 𝑌 ∩ 𝐴 ) = 𝐴 )
2	1	bilani	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → ( 𝑌 ∩ 𝐴 ) = 𝐴 )
3		simpl	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐿 ∈ ( Fil ‘ 𝑌 ) )
4		id	⊢ ( 𝐴 ⊆ 𝑌 → 𝐴 ⊆ 𝑌 )
5		filtop	⊢ ( 𝐿 ∈ ( Fil ‘ 𝑌 ) → 𝑌 ∈ 𝐿 )
6		ssexg	⊢ ( ( 𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝐿 ) → 𝐴 ∈ V )
7	4 5 6	syl2anr	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 ∈ V )
8	5	adantr	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝑌 ∈ 𝐿 )
9		elrestr	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ∈ V ∧ 𝑌 ∈ 𝐿 ) → ( 𝑌 ∩ 𝐴 ) ∈ ( 𝐿 ↾t 𝐴 ) )
10	3 7 8 9	syl3anc	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → ( 𝑌 ∩ 𝐴 ) ∈ ( 𝐿 ↾t 𝐴 ) )
11	2 10	eqeltrrd	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 ∈ ( 𝐿 ↾t 𝐴 ) )
12		elssuni	⊢ ( 𝐴 ∈ ( 𝐿 ↾t 𝐴 ) → 𝐴 ⊆ ∪ ( 𝐿 ↾t 𝐴 ) )
13	11 12	syl	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 ⊆ ∪ ( 𝐿 ↾t 𝐴 ) )
14		restsspw	⊢ ( 𝐿 ↾t 𝐴 ) ⊆ 𝒫 𝐴
15		sspwuni	⊢ ( ( 𝐿 ↾t 𝐴 ) ⊆ 𝒫 𝐴 ↔ ∪ ( 𝐿 ↾t 𝐴 ) ⊆ 𝐴 )
16	14 15	mpbi	⊢ ∪ ( 𝐿 ↾t 𝐴 ) ⊆ 𝐴
17	16	a1i	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → ∪ ( 𝐿 ↾t 𝐴 ) ⊆ 𝐴 )
18	13 17	eqssd	⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 = ∪ ( 𝐿 ↾t 𝐴 ) )