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Theorem trfil2 21501
 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.)
Assertion
Ref Expression
trfil2 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣𝐿 (𝑣𝐴) ≠ ∅))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐿   𝑣,𝑌

Proof of Theorem trfil2
Dummy variables 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐴𝑌)
2 sseqin2 3779 . . . . 5 (𝐴𝑌 ↔ (𝑌𝐴) = 𝐴)
31, 2sylib 207 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑌𝐴) = 𝐴)
4 simpl 472 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐿 ∈ (Fil‘𝑌))
5 id 22 . . . . . 6 (𝐴𝑌𝐴𝑌)
6 filtop 21469 . . . . . 6 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
7 ssexg 4732 . . . . . 6 ((𝐴𝑌𝑌𝐿) → 𝐴 ∈ V)
85, 6, 7syl2anr 494 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐴 ∈ V)
96adantr 480 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝑌𝐿)
10 elrestr 15912 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ 𝑌𝐿) → (𝑌𝐴) ∈ (𝐿t 𝐴))
114, 8, 9, 10syl3anc 1318 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑌𝐴) ∈ (𝐿t 𝐴))
123, 11eqeltrrd 2689 . . 3 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐴 ∈ (𝐿t 𝐴))
13 elpwi 4117 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
14 vex 3176 . . . . . . . . . 10 𝑢 ∈ V
1514inex1 4727 . . . . . . . . 9 (𝑢𝐴) ∈ V
1615a1i 11 . . . . . . . 8 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ 𝑢𝐿) → (𝑢𝐴) ∈ V)
17 elrest 15911 . . . . . . . . . 10 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (𝑦 ∈ (𝐿t 𝐴) ↔ ∃𝑢𝐿 𝑦 = (𝑢𝐴)))
188, 17syldan 486 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑦 ∈ (𝐿t 𝐴) ↔ ∃𝑢𝐿 𝑦 = (𝑢𝐴)))
1918adantr 480 . . . . . . . 8 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) → (𝑦 ∈ (𝐿t 𝐴) ↔ ∃𝑢𝐿 𝑦 = (𝑢𝐴)))
20 simpr 476 . . . . . . . . 9 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ 𝑦 = (𝑢𝐴)) → 𝑦 = (𝑢𝐴))
2120sseq1d 3595 . . . . . . . 8 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ 𝑦 = (𝑢𝐴)) → (𝑦𝑥 ↔ (𝑢𝐴) ⊆ 𝑥))
2216, 19, 21rexxfr2d 4809 . . . . . . 7 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) → (∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥 ↔ ∃𝑢𝐿 (𝑢𝐴) ⊆ 𝑥))
23 indir 3834 . . . . . . . . . 10 ((𝑢𝑥) ∩ 𝐴) = ((𝑢𝐴) ∪ (𝑥𝐴))
24 simplr 788 . . . . . . . . . . . . 13 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑥𝐴)
25 df-ss 3554 . . . . . . . . . . . . 13 (𝑥𝐴 ↔ (𝑥𝐴) = 𝑥)
2624, 25sylib 207 . . . . . . . . . . . 12 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → (𝑥𝐴) = 𝑥)
2726uneq2d 3729 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → ((𝑢𝐴) ∪ (𝑥𝐴)) = ((𝑢𝐴) ∪ 𝑥))
28 simprr 792 . . . . . . . . . . . 12 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → (𝑢𝐴) ⊆ 𝑥)
29 ssequn1 3745 . . . . . . . . . . . 12 ((𝑢𝐴) ⊆ 𝑥 ↔ ((𝑢𝐴) ∪ 𝑥) = 𝑥)
3028, 29sylib 207 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → ((𝑢𝐴) ∪ 𝑥) = 𝑥)
3127, 30eqtrd 2644 . . . . . . . . . 10 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → ((𝑢𝐴) ∪ (𝑥𝐴)) = 𝑥)
3223, 31syl5eq 2656 . . . . . . . . 9 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → ((𝑢𝑥) ∩ 𝐴) = 𝑥)
33 simplll 794 . . . . . . . . . 10 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝐿 ∈ (Fil‘𝑌))
34 simpllr 795 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝐴𝑌)
3533, 34, 8syl2anc 691 . . . . . . . . . 10 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝐴 ∈ V)
36 simprl 790 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑢𝐿)
37 filelss 21466 . . . . . . . . . . . . 13 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑢𝐿) → 𝑢𝑌)
3833, 36, 37syl2anc 691 . . . . . . . . . . . 12 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑢𝑌)
3924, 34sstrd 3578 . . . . . . . . . . . 12 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑥𝑌)
4038, 39unssd 3751 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → (𝑢𝑥) ⊆ 𝑌)
41 ssun1 3738 . . . . . . . . . . . 12 𝑢 ⊆ (𝑢𝑥)
4241a1i 11 . . . . . . . . . . 11 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑢 ⊆ (𝑢𝑥))
43 filss 21467 . . . . . . . . . . 11 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑢𝐿 ∧ (𝑢𝑥) ⊆ 𝑌𝑢 ⊆ (𝑢𝑥))) → (𝑢𝑥) ∈ 𝐿)
4433, 36, 40, 42, 43syl13anc 1320 . . . . . . . . . 10 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → (𝑢𝑥) ∈ 𝐿)
45 elrestr 15912 . . . . . . . . . 10 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ (𝑢𝑥) ∈ 𝐿) → ((𝑢𝑥) ∩ 𝐴) ∈ (𝐿t 𝐴))
4633, 35, 44, 45syl3anc 1318 . . . . . . . . 9 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → ((𝑢𝑥) ∩ 𝐴) ∈ (𝐿t 𝐴))
4732, 46eqeltrrd 2689 . . . . . . . 8 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) ∧ (𝑢𝐿 ∧ (𝑢𝐴) ⊆ 𝑥)) → 𝑥 ∈ (𝐿t 𝐴))
4847rexlimdvaa 3014 . . . . . . 7 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) → (∃𝑢𝐿 (𝑢𝐴) ⊆ 𝑥𝑥 ∈ (𝐿t 𝐴)))
4922, 48sylbid 229 . . . . . 6 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥𝐴) → (∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)))
5049ex 449 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑥𝐴 → (∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴))))
5113, 50syl5 33 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑥 ∈ 𝒫 𝐴 → (∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴))))
5251ralrimiv 2948 . . 3 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)))
53 simpll 786 . . . . . 6 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝐿𝑢𝐿)) → 𝐿 ∈ (Fil‘𝑌))
548adantr 480 . . . . . 6 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝐿𝑢𝐿)) → 𝐴 ∈ V)
55 filin 21468 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑧𝐿𝑢𝐿) → (𝑧𝑢) ∈ 𝐿)
56553expb 1258 . . . . . . 7 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑧𝐿𝑢𝐿)) → (𝑧𝑢) ∈ 𝐿)
5756adantlr 747 . . . . . 6 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝐿𝑢𝐿)) → (𝑧𝑢) ∈ 𝐿)
58 elrestr 15912 . . . . . 6 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ (𝑧𝑢) ∈ 𝐿) → ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴))
5953, 54, 57, 58syl3anc 1318 . . . . 5 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝐿𝑢𝐿)) → ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴))
6059ralrimivva 2954 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ∀𝑧𝐿𝑢𝐿 ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴))
61 vex 3176 . . . . . . 7 𝑧 ∈ V
6261inex1 4727 . . . . . 6 (𝑧𝐴) ∈ V
6362a1i 11 . . . . 5 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝐿) → (𝑧𝐴) ∈ V)
64 elrest 15911 . . . . . 6 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐿t 𝐴) ↔ ∃𝑧𝐿 𝑥 = (𝑧𝐴)))
658, 64syldan 486 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (𝑥 ∈ (𝐿t 𝐴) ↔ ∃𝑧𝐿 𝑥 = (𝑧𝐴)))
6615a1i 11 . . . . . 6 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥 = (𝑧𝐴)) ∧ 𝑢𝐿) → (𝑢𝐴) ∈ V)
6718adantr 480 . . . . . 6 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥 = (𝑧𝐴)) → (𝑦 ∈ (𝐿t 𝐴) ↔ ∃𝑢𝐿 𝑦 = (𝑢𝐴)))
68 ineq12 3771 . . . . . . . . 9 ((𝑥 = (𝑧𝐴) ∧ 𝑦 = (𝑢𝐴)) → (𝑥𝑦) = ((𝑧𝐴) ∩ (𝑢𝐴)))
69 inindir 3793 . . . . . . . . 9 ((𝑧𝑢) ∩ 𝐴) = ((𝑧𝐴) ∩ (𝑢𝐴))
7068, 69syl6eqr 2662 . . . . . . . 8 ((𝑥 = (𝑧𝐴) ∧ 𝑦 = (𝑢𝐴)) → (𝑥𝑦) = ((𝑧𝑢) ∩ 𝐴))
7170adantll 746 . . . . . . 7 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥 = (𝑧𝐴)) ∧ 𝑦 = (𝑢𝐴)) → (𝑥𝑦) = ((𝑧𝑢) ∩ 𝐴))
7271eleq1d 2672 . . . . . 6 ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥 = (𝑧𝐴)) ∧ 𝑦 = (𝑢𝐴)) → ((𝑥𝑦) ∈ (𝐿t 𝐴) ↔ ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴)))
7366, 67, 72ralxfr2d 4808 . . . . 5 (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) ∧ 𝑥 = (𝑧𝐴)) → (∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴) ↔ ∀𝑢𝐿 ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴)))
7463, 65, 73ralxfr2d 4808 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴) ↔ ∀𝑧𝐿𝑢𝐿 ((𝑧𝑢) ∩ 𝐴) ∈ (𝐿t 𝐴)))
7560, 74mpbird 246 . . 3 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))
76 isfil2 21470 . . . . . 6 ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ (((𝐿t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)))
77 restsspw 15915 . . . . . . . 8 (𝐿t 𝐴) ⊆ 𝒫 𝐴
78 3anass 1035 . . . . . . . 8 (((𝐿t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ↔ ((𝐿t 𝐴) ⊆ 𝒫 𝐴 ∧ (¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴))))
7977, 78mpbiran 955 . . . . . . 7 (((𝐿t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ↔ (¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)))
80793anbi1i 1246 . . . . . 6 ((((𝐿t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)) ↔ ((¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)))
81 3anass 1035 . . . . . 6 (((¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)) ↔ ((¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))))
8276, 80, 813bitri 285 . . . . 5 ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ((¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))))
83 anass 679 . . . . 5 (((¬ ∅ ∈ (𝐿t 𝐴) ∧ 𝐴 ∈ (𝐿t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))) ↔ (¬ ∅ ∈ (𝐿t 𝐴) ∧ (𝐴 ∈ (𝐿t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)))))
84 ancom 465 . . . . 5 ((¬ ∅ ∈ (𝐿t 𝐴) ∧ (𝐴 ∈ (𝐿t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴)))) ↔ ((𝐴 ∈ (𝐿t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))) ∧ ¬ ∅ ∈ (𝐿t 𝐴)))
8582, 83, 843bitri 285 . . . 4 ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ((𝐴 ∈ (𝐿t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))) ∧ ¬ ∅ ∈ (𝐿t 𝐴)))
8685baib 942 . . 3 ((𝐴 ∈ (𝐿t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿t 𝐴)𝑦𝑥𝑥 ∈ (𝐿t 𝐴)) ∧ ∀𝑥 ∈ (𝐿t 𝐴)∀𝑦 ∈ (𝐿t 𝐴)(𝑥𝑦) ∈ (𝐿t 𝐴))) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ ∅ ∈ (𝐿t 𝐴)))
8712, 52, 75, 86syl12anc 1316 . 2 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ ∅ ∈ (𝐿t 𝐴)))
88 nesym 2838 . . . 4 ((𝑣𝐴) ≠ ∅ ↔ ¬ ∅ = (𝑣𝐴))
8988ralbii 2963 . . 3 (∀𝑣𝐿 (𝑣𝐴) ≠ ∅ ↔ ∀𝑣𝐿 ¬ ∅ = (𝑣𝐴))
90 elrest 15911 . . . . . 6 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (∅ ∈ (𝐿t 𝐴) ↔ ∃𝑣𝐿 ∅ = (𝑣𝐴)))
918, 90syldan 486 . . . . 5 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (∅ ∈ (𝐿t 𝐴) ↔ ∃𝑣𝐿 ∅ = (𝑣𝐴)))
92 dfrex2 2979 . . . . 5 (∃𝑣𝐿 ∅ = (𝑣𝐴) ↔ ¬ ∀𝑣𝐿 ¬ ∅ = (𝑣𝐴))
9391, 92syl6bb 275 . . . 4 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (∅ ∈ (𝐿t 𝐴) ↔ ¬ ∀𝑣𝐿 ¬ ∅ = (𝑣𝐴)))
9493con2bid 343 . . 3 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (∀𝑣𝐿 ¬ ∅ = (𝑣𝐴) ↔ ¬ ∅ ∈ (𝐿t 𝐴)))
9589, 94syl5bb 271 . 2 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → (∀𝑣𝐿 (𝑣𝐴) ≠ ∅ ↔ ¬ ∅ ∈ (𝐿t 𝐴)))
9687, 95bitr4d 270 1 ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣𝐿 (𝑣𝐴) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Filcfil 21459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-fbas 19564  df-fil 21460 This theorem is referenced by:  trfil3  21502  trnei  21506
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