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Theorem tailfb 32687
Description: The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypothesis
Ref Expression
tailfb.1 𝑋 = dom 𝐷
Assertion
Ref Expression
tailfb ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋))

Proof of Theorem tailfb
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tailfb.1 . . . . 5 𝑋 = dom 𝐷
21tailf 32685 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
32frnd 6257 . . 3 (𝐷 ∈ DirRel → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
43adantr 468 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
5 n0 4126 . . . . 5 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
6 ffn 6250 . . . . . . . 8 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → (tail‘𝐷) Fn 𝑋)
7 fnfvelrn 6572 . . . . . . . . 9 (((tail‘𝐷) Fn 𝑋𝑥𝑋) → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷))
87ex 399 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥𝑋 → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷)))
92, 6, 83syl 18 . . . . . . 7 (𝐷 ∈ DirRel → (𝑥𝑋 → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷)))
10 ne0i 4116 . . . . . . 7 (((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷) → ran (tail‘𝐷) ≠ ∅)
119, 10syl6 35 . . . . . 6 (𝐷 ∈ DirRel → (𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
1211exlimdv 2023 . . . . 5 (𝐷 ∈ DirRel → (∃𝑥 𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
135, 12syl5bi 233 . . . 4 (𝐷 ∈ DirRel → (𝑋 ≠ ∅ → ran (tail‘𝐷) ≠ ∅))
1413imp 395 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ≠ ∅)
151tailini 32686 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → 𝑥 ∈ ((tail‘𝐷)‘𝑥))
16 n0i 4115 . . . . . . . 8 (𝑥 ∈ ((tail‘𝐷)‘𝑥) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1715, 16syl 17 . . . . . . 7 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1817nrexdv 3184 . . . . . 6 (𝐷 ∈ DirRel → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
1918adantr 468 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
20 fvelrnb 6458 . . . . . . 7 ((tail‘𝐷) Fn 𝑋 → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
212, 6, 203syl 18 . . . . . 6 (𝐷 ∈ DirRel → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
2221adantr 468 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
2319, 22mtbird 316 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∅ ∈ ran (tail‘𝐷))
24 df-nel 3078 . . . 4 (∅ ∉ ran (tail‘𝐷) ↔ ¬ ∅ ∈ ran (tail‘𝐷))
2523, 24sylibr 225 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∅ ∉ ran (tail‘𝐷))
26 fvelrnb 6458 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥 ∈ ran (tail‘𝐷) ↔ ∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥))
27 fvelrnb 6458 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑦 ∈ ran (tail‘𝐷) ↔ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
2826, 27anbi12d 618 . . . . . . 7 ((tail‘𝐷) Fn 𝑋 → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦)))
292, 6, 283syl 18 . . . . . 6 (𝐷 ∈ DirRel → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦)))
30 reeanv 3291 . . . . . . 7 (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
311dirge 17436 . . . . . . . . . . 11 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑣𝑋) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
32313expb 1142 . . . . . . . . . 10 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
332, 6syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷) Fn 𝑋)
34 fnfvelrn 6572 . . . . . . . . . . . . 13 (((tail‘𝐷) Fn 𝑋𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3533, 34sylan 571 . . . . . . . . . . . 12 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3635ad2ant2r 744 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
37 dirtr 17435 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑢𝐷𝑤𝑤𝐷𝑥)) → 𝑢𝐷𝑥)
3837exp32 409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
3938elvd 3392 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4039com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑢𝐷𝑤𝑢𝐷𝑥)))
4140imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑢𝐷𝑤𝑢𝐷𝑥))
4241ad2ant2rl 746 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑢𝐷𝑤𝑢𝐷𝑥))
43 dirtr 17435 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑣𝐷𝑤𝑤𝐷𝑥)) → 𝑣𝐷𝑥)
4443exp32 409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4544elvd 3392 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4645com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑣𝐷𝑤𝑣𝐷𝑥)))
4746imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑣𝐷𝑤𝑣𝐷𝑥))
4847ad2ant2rl 746 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑣𝐷𝑤𝑣𝐷𝑥))
4942, 48anim12d 598 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥)))
5049expr 446 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → (𝑤𝐷𝑥 → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5150com23 86 . . . . . . . . . . . . . . 15 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5251impr 444 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥)))
53 vex 3390 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
541eltail 32684 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑤𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5553, 54mp3an3 1567 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5655ad2ant2r 744 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
571eltail 32684 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
5853, 57mp3an3 1567 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑢𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
5958adantrr 699 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
601eltail 32684 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑣𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6153, 60mp3an3 1567 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑣𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6261adantrl 698 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6359, 62anbi12d 618 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)) ↔ (𝑢𝐷𝑥𝑣𝐷𝑥)))
6463adantr 468 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)) ↔ (𝑢𝐷𝑥𝑣𝐷𝑥)))
6552, 56, 643imtr4d 285 . . . . . . . . . . . . 13 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣))))
66 elin 3989 . . . . . . . . . . . . 13 (𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)))
6765, 66syl6ibr 243 . . . . . . . . . . . 12 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → 𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
6867ssrdv 3798 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
69 sseq1 3817 . . . . . . . . . . . 12 (𝑧 = ((tail‘𝐷)‘𝑤) → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
7069rspcev 3498 . . . . . . . . . . 11 ((((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷) ∧ ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
7136, 68, 70syl2anc 575 . . . . . . . . . 10 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
7232, 71rexlimddv 3219 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
73 ineq1 4000 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑢) = 𝑥 → (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) = (𝑥 ∩ ((tail‘𝐷)‘𝑣)))
7473sseq2d 3824 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑢) = 𝑥 → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣))))
7574rexbidv 3236 . . . . . . . . . 10 (((tail‘𝐷)‘𝑢) = 𝑥 → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣))))
76 ineq2 4001 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑥 ∩ ((tail‘𝐷)‘𝑣)) = (𝑥𝑦))
7776sseq2d 3824 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥𝑦)))
7877rexbidv 3236 . . . . . . . . . 10 (((tail‘𝐷)‘𝑣) = 𝑦 → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
7975, 78sylan9bb 501 . . . . . . . . 9 ((((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8072, 79syl5ibcom 236 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8180rexlimdvva 3222 . . . . . . 7 (𝐷 ∈ DirRel → (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8230, 81syl5bir 234 . . . . . 6 (𝐷 ∈ DirRel → ((∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8329, 82sylbid 231 . . . . 5 (𝐷 ∈ DirRel → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8483adantr 468 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8584ralrimivv 3154 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦))
8614, 25, 853jca 1151 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (ran (tail‘𝐷) ≠ ∅ ∧ ∅ ∉ ran (tail‘𝐷) ∧ ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
87 dmexg 7321 . . . . 5 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
881, 87syl5eqel 2885 . . . 4 (𝐷 ∈ DirRel → 𝑋 ∈ V)
8988adantr 468 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V)
90 isfbas2 21846 . . 3 (𝑋 ∈ V → (ran (tail‘𝐷) ∈ (fBas‘𝑋) ↔ (ran (tail‘𝐷) ⊆ 𝒫 𝑋 ∧ (ran (tail‘𝐷) ≠ ∅ ∧ ∅ ∉ ran (tail‘𝐷) ∧ ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))))
9189, 90syl 17 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (ran (tail‘𝐷) ∈ (fBas‘𝑋) ↔ (ran (tail‘𝐷) ⊆ 𝒫 𝑋 ∧ (ran (tail‘𝐷) ≠ ∅ ∧ ∅ ∉ ran (tail‘𝐷) ∧ ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))))
924, 86, 91mpbir2and 695 1 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2155  wne 2974  wnel 3077  wral 3092  wrex 3093  Vcvv 3387  cin 3762  wss 3763  c0 4110  𝒫 cpw 4345   class class class wbr 4837  dom cdm 5305  ran crn 5306   Fn wfn 6090  wf 6091  cfv 6095  DirRelcdir 17427  tailctail 17428  fBascfbas 19936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-dir 17429  df-tail 17430  df-fbas 19945
This theorem is referenced by:  filnetlem4  32691
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