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Theorem tailfb 32676
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 32674 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
3 frn 6212 . . . 4 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
42, 3syl 17 . . 3 (𝐷 ∈ DirRel → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
54adantr 472 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
6 n0 4072 . . . . 5 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
7 ffn 6204 . . . . . . . 8 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → (tail‘𝐷) Fn 𝑋)
8 fnfvelrn 6517 . . . . . . . . 9 (((tail‘𝐷) Fn 𝑋𝑥𝑋) → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷))
98ex 449 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥𝑋 → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷)))
102, 7, 93syl 18 . . . . . . 7 (𝐷 ∈ DirRel → (𝑥𝑋 → ((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷)))
11 ne0i 4062 . . . . . . 7 (((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷) → ran (tail‘𝐷) ≠ ∅)
1210, 11syl6 35 . . . . . 6 (𝐷 ∈ DirRel → (𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
1312exlimdv 2008 . . . . 5 (𝐷 ∈ DirRel → (∃𝑥 𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
146, 13syl5bi 232 . . . 4 (𝐷 ∈ DirRel → (𝑋 ≠ ∅ → ran (tail‘𝐷) ≠ ∅))
1514imp 444 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ≠ ∅)
161tailini 32675 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → 𝑥 ∈ ((tail‘𝐷)‘𝑥))
17 n0i 4061 . . . . . . . 8 (𝑥 ∈ ((tail‘𝐷)‘𝑥) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1816, 17syl 17 . . . . . . 7 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1918nrexdv 3137 . . . . . 6 (𝐷 ∈ DirRel → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
2019adantr 472 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
21 fvelrnb 6403 . . . . . . 7 ((tail‘𝐷) Fn 𝑋 → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
222, 7, 213syl 18 . . . . . 6 (𝐷 ∈ DirRel → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
2322adantr 472 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
2420, 23mtbird 314 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∅ ∈ ran (tail‘𝐷))
25 df-nel 3034 . . . 4 (∅ ∉ ran (tail‘𝐷) ↔ ¬ ∅ ∈ ran (tail‘𝐷))
2624, 25sylibr 224 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∅ ∉ ran (tail‘𝐷))
27 fvelrnb 6403 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥 ∈ ran (tail‘𝐷) ↔ ∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥))
28 fvelrnb 6403 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑦 ∈ ran (tail‘𝐷) ↔ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
2927, 28anbi12d 749 . . . . . . 7 ((tail‘𝐷) Fn 𝑋 → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦)))
302, 7, 293syl 18 . . . . . 6 (𝐷 ∈ DirRel → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦)))
31 reeanv 3243 . . . . . . 7 (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
321dirge 17436 . . . . . . . . . . 11 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑣𝑋) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
33323expb 1114 . . . . . . . . . 10 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
342, 7syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷) Fn 𝑋)
35 fnfvelrn 6517 . . . . . . . . . . . . 13 (((tail‘𝐷) Fn 𝑋𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3634, 35sylan 489 . . . . . . . . . . . 12 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3736ad2ant2r 800 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
38 vex 3341 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
39 dirtr 17435 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑢𝐷𝑤𝑤𝐷𝑥)) → 𝑢𝐷𝑥)
4039exp32 632 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4138, 40mpan2 709 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4241com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑢𝐷𝑤𝑢𝐷𝑥)))
4342imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑢𝐷𝑤𝑢𝐷𝑥))
4443ad2ant2rl 802 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑢𝐷𝑤𝑢𝐷𝑥))
45 dirtr 17435 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑣𝐷𝑤𝑤𝐷𝑥)) → 𝑣𝐷𝑥)
4645exp32 632 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4738, 46mpan2 709 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4847com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑣𝐷𝑤𝑣𝐷𝑥)))
4948imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5049ad2ant2rl 802 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5144, 50anim12d 587 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥)))
5251expr 644 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → (𝑤𝐷𝑥 → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5352com23 86 . . . . . . . . . . . . . . 15 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5453impr 650 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥)))
551eltail 32673 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑤𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5638, 55mp3an3 1560 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5756ad2ant2r 800 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
581eltail 32673 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
5938, 58mp3an3 1560 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑢𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
6059adantrr 755 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
611eltail 32673 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑣𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6238, 61mp3an3 1560 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑣𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6362adantrl 754 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6460, 63anbi12d 749 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)) ↔ (𝑢𝐷𝑥𝑣𝐷𝑥)))
6564adantr 472 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)) ↔ (𝑢𝐷𝑥𝑣𝐷𝑥)))
6654, 57, 653imtr4d 283 . . . . . . . . . . . . 13 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣))))
67 elin 3937 . . . . . . . . . . . . 13 (𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)))
6866, 67syl6ibr 242 . . . . . . . . . . . 12 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → 𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
6968ssrdv 3748 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
70 sseq1 3765 . . . . . . . . . . . 12 (𝑧 = ((tail‘𝐷)‘𝑤) → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
7170rspcev 3447 . . . . . . . . . . 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)))
7237, 69, 71syl2anc 696 . . . . . . . . . 10 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
7333, 72rexlimddv 3171 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
74 ineq1 3948 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑢) = 𝑥 → (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) = (𝑥 ∩ ((tail‘𝐷)‘𝑣)))
7574sseq2d 3772 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑢) = 𝑥 → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣))))
7675rexbidv 3188 . . . . . . . . . 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))))
77 ineq2 3949 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑥 ∩ ((tail‘𝐷)‘𝑣)) = (𝑥𝑦))
7877sseq2d 3772 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥𝑦)))
7978rexbidv 3188 . . . . . . . . . 10 (((tail‘𝐷)‘𝑣) = 𝑦 → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8076, 79sylan9bb 738 . . . . . . . . 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)))
8173, 80syl5ibcom 235 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8281rexlimdvva 3174 . . . . . . 7 (𝐷 ∈ DirRel → (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8331, 82syl5bir 233 . . . . . 6 (𝐷 ∈ DirRel → ((∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8430, 83sylbid 230 . . . . 5 (𝐷 ∈ DirRel → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8584adantr 472 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8685ralrimivv 3106 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦))
8715, 26, 863jca 1123 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (ran (tail‘𝐷) ≠ ∅ ∧ ∅ ∉ ran (tail‘𝐷) ∧ ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
88 dmexg 7260 . . . . 5 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
891, 88syl5eqel 2841 . . . 4 (𝐷 ∈ DirRel → 𝑋 ∈ V)
9089adantr 472 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V)
91 isfbas2 21838 . . 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)))))
9290, 91syl 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)))))
935, 87, 92mpbir2and 995 1 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wex 1851  wcel 2137  wne 2930  wnel 3033  wral 3048  wrex 3049  Vcvv 3338  cin 3712  wss 3713  c0 4056  𝒫 cpw 4300   class class class wbr 4802  dom cdm 5264  ran crn 5265   Fn wfn 6042  wf 6043  cfv 6047  DirRelcdir 17427  tailctail 17428  fBascfbas 19934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-dir 17429  df-tail 17430  df-fbas 19943
This theorem is referenced by:  filnetlem4  32680
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