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Theorem tailfb 32497
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 32495 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
3 frn 6091 . . . 4 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
42, 3syl 17 . . 3 (𝐷 ∈ DirRel → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
54adantr 480 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ⊆ 𝒫 𝑋)
6 n0 3964 . . . . 5 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
7 ffn 6083 . . . . . . . 8 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → (tail‘𝐷) Fn 𝑋)
8 fnfvelrn 6396 . . . . . . . . 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 3954 . . . . . . 7 (((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷) → ran (tail‘𝐷) ≠ ∅)
1210, 11syl6 35 . . . . . 6 (𝐷 ∈ DirRel → (𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
1312exlimdv 1901 . . . . 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 32496 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → 𝑥 ∈ ((tail‘𝐷)‘𝑥))
17 n0i 3953 . . . . . . . 8 (𝑥 ∈ ((tail‘𝐷)‘𝑥) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1816, 17syl 17 . . . . . . 7 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1918nrexdv 3030 . . . . . 6 (𝐷 ∈ DirRel → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
2019adantr 480 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
21 fvelrnb 6282 . . . . . . 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 480 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → (∅ ∈ ran (tail‘𝐷) ↔ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅))
2420, 23mtbird 314 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∅ ∈ ran (tail‘𝐷))
25 df-nel 2927 . . . 4 (∅ ∉ ran (tail‘𝐷) ↔ ¬ ∅ ∈ ran (tail‘𝐷))
2624, 25sylibr 224 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∅ ∉ ran (tail‘𝐷))
27 fvelrnb 6282 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥 ∈ ran (tail‘𝐷) ↔ ∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥))
28 fvelrnb 6282 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑦 ∈ ran (tail‘𝐷) ↔ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
2927, 28anbi12d 747 . . . . . . 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 3136 . . . . . . 7 (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
321dirge 17284 . . . . . . . . . . 11 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑣𝑋) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
33323expb 1285 . . . . . . . . . 10 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
342, 7syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷) Fn 𝑋)
35 fnfvelrn 6396 . . . . . . . . . . . . 13 (((tail‘𝐷) Fn 𝑋𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3634, 35sylan 487 . . . . . . . . . . . 12 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
3736ad2ant2r 798 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
38 vex 3234 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
39 dirtr 17283 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑢𝐷𝑤𝑤𝐷𝑥)) → 𝑢𝐷𝑥)
4039exp32 630 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4138, 40mpan2 707 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4241com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑢𝐷𝑤𝑢𝐷𝑥)))
4342imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑢𝐷𝑤𝑢𝐷𝑥))
4443ad2ant2rl 800 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑢𝐷𝑤𝑢𝐷𝑥))
45 dirtr 17283 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑣𝐷𝑤𝑤𝐷𝑥)) → 𝑣𝐷𝑥)
4645exp32 630 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4738, 46mpan2 707 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4847com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑣𝐷𝑤𝑣𝐷𝑥)))
4948imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5049ad2ant2rl 800 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5144, 50anim12d 585 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥)))
5251expr 642 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → (𝑤𝐷𝑥 → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5352com23 86 . . . . . . . . . . . . . . 15 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5453impr 648 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥)))
551eltail 32494 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑤𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5638, 55mp3an3 1453 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5756ad2ant2r 798 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
581eltail 32494 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
5938, 58mp3an3 1453 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑢𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
6059adantrr 753 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
611eltail 32494 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑣𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6238, 61mp3an3 1453 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑣𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6362adantrl 752 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6460, 63anbi12d 747 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)) ↔ (𝑢𝐷𝑥𝑣𝐷𝑥)))
6564adantr 480 . . . . . . . . . . . . . 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 3829 . . . . . . . . . . . . 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 3642 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
70 sseq1 3659 . . . . . . . . . . . 12 (𝑧 = ((tail‘𝐷)‘𝑤) → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
7170rspcev 3340 . . . . . . . . . . 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 694 . . . . . . . . . 10 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
7333, 72rexlimddv 3064 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
74 ineq1 3840 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑢) = 𝑥 → (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) = (𝑥 ∩ ((tail‘𝐷)‘𝑣)))
7574sseq2d 3666 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑢) = 𝑥 → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣))))
7675rexbidv 3081 . . . . . . . . . 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 3841 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑥 ∩ ((tail‘𝐷)‘𝑣)) = (𝑥𝑦))
7877sseq2d 3666 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥𝑦)))
7978rexbidv 3081 . . . . . . . . . 10 (((tail‘𝐷)‘𝑣) = 𝑦 → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8076, 79sylan9bb 736 . . . . . . . . 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 3067 . . . . . . 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 480 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ((𝑥 ∈ ran (tail‘𝐷) ∧ 𝑦 ∈ ran (tail‘𝐷)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8685ralrimivv 2999 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦))
8715, 26, 863jca 1261 . 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 7139 . . . . 5 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
891, 88syl5eqel 2734 . . . 4 (𝐷 ∈ DirRel → 𝑋 ∈ V)
9089adantr 480 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V)
91 isfbas2 21686 . . 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 977 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 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wnel 2926  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  DirRelcdir 17275  tailctail 17276  fBascfbas 19782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-dir 17277  df-tail 17278  df-fbas 19791
This theorem is referenced by:  filnetlem4  32501
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