Mathbox for Jeff Hankins < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tailfb Structured version   Visualization version   GIF version

Theorem tailfb 31542
 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 31540 . . . 4 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
3 frn 5966 . . . 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 3890 . . . . 5 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
7 ffn 5958 . . . . . . . 8 ((tail‘𝐷):𝑋⟶𝒫 𝑋 → (tail‘𝐷) Fn 𝑋)
8 fnfvelrn 6264 . . . . . . . . 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 3880 . . . . . . 7 (((tail‘𝐷)‘𝑥) ∈ ran (tail‘𝐷) → ran (tail‘𝐷) ≠ ∅)
1210, 11syl6 34 . . . . . 6 (𝐷 ∈ DirRel → (𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
1312exlimdv 1848 . . . . 5 (𝐷 ∈ DirRel → (∃𝑥 𝑥𝑋 → ran (tail‘𝐷) ≠ ∅))
146, 13syl5bi 231 . . . 4 (𝐷 ∈ DirRel → (𝑋 ≠ ∅ → ran (tail‘𝐷) ≠ ∅))
1514imp 444 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ≠ ∅)
161tailini 31541 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → 𝑥 ∈ ((tail‘𝐷)‘𝑥))
17 n0i 3879 . . . . . . . 8 (𝑥 ∈ ((tail‘𝐷)‘𝑥) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1816, 17syl 17 . . . . . . 7 ((𝐷 ∈ DirRel ∧ 𝑥𝑋) → ¬ ((tail‘𝐷)‘𝑥) = ∅)
1918nrexdv 2984 . . . . . 6 (𝐷 ∈ DirRel → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
2019adantr 480 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ¬ ∃𝑥𝑋 ((tail‘𝐷)‘𝑥) = ∅)
21 fvelrnb 6153 . . . . . . 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 2783 . . . 4 (∅ ∉ ran (tail‘𝐷) ↔ ¬ ∅ ∈ ran (tail‘𝐷))
2624, 25sylibr 223 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∅ ∉ ran (tail‘𝐷))
27 fvelrnb 6153 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑥 ∈ ran (tail‘𝐷) ↔ ∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥))
28 fvelrnb 6153 . . . . . . . 8 ((tail‘𝐷) Fn 𝑋 → (𝑦 ∈ ran (tail‘𝐷) ↔ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
2927, 28anbi12d 743 . . . . . . 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 3086 . . . . . . 7 (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) ↔ (∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦))
321dirge 17060 . . . . . . . . . . 11 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑣𝑋) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
33323expb 1258 . . . . . . . . . 10 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑤𝑋 (𝑢𝐷𝑤𝑣𝐷𝑤))
342, 7syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel → (tail‘𝐷) Fn 𝑋)
35 fnfvelrn 6264 . . . . . . . . . . . . 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 779 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ∈ ran (tail‘𝐷))
38 vex 3176 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
39 dirtr 17059 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑢𝐷𝑤𝑤𝐷𝑥)) → 𝑢𝐷𝑥)
4039exp32 629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4138, 40mpan2 703 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑢𝐷𝑤 → (𝑤𝐷𝑥𝑢𝐷𝑥)))
4241com23 84 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑢𝐷𝑤𝑢𝐷𝑥)))
4342imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑢𝐷𝑤𝑢𝐷𝑥))
4443ad2ant2rl 781 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑢𝐷𝑤𝑢𝐷𝑥))
45 dirtr 17059 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) ∧ (𝑣𝐷𝑤𝑤𝐷𝑥)) → 𝑣𝐷𝑥)
4645exp32 629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ 𝑥 ∈ V) → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4738, 46mpan2 703 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel → (𝑣𝐷𝑤 → (𝑤𝐷𝑥𝑣𝐷𝑥)))
4847com23 84 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel → (𝑤𝐷𝑥 → (𝑣𝐷𝑤𝑣𝐷𝑥)))
4948imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑤𝐷𝑥) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5049ad2ant2rl 781 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → (𝑣𝐷𝑤𝑣𝐷𝑥))
5144, 50anim12d 584 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋𝑤𝐷𝑥)) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥)))
5251expr 641 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → (𝑤𝐷𝑥 → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5352com23 84 . . . . . . . . . . . . . . 15 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ 𝑤𝑋) → ((𝑢𝐷𝑤𝑣𝐷𝑤) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥))))
5453impr 647 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑤𝐷𝑥 → (𝑢𝐷𝑥𝑣𝐷𝑥)))
551eltail 31539 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑤𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5638, 55mp3an3 1405 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ 𝑤𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
5756ad2ant2r 779 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) ↔ 𝑤𝐷𝑥))
581eltail 31539 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑢𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
5938, 58mp3an3 1405 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑢𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
6059adantrr 749 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ↔ 𝑢𝐷𝑥))
611eltail 31539 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑣𝑋𝑥 ∈ V) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6238, 61mp3an3 1405 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑣𝑋) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6362adantrl 748 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → (𝑥 ∈ ((tail‘𝐷)‘𝑣) ↔ 𝑣𝐷𝑥))
6460, 63anbi12d 743 . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . 13 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣))))
67 elin 3758 . . . . . . . . . . . . 13 (𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ (𝑥 ∈ ((tail‘𝐷)‘𝑢) ∧ 𝑥 ∈ ((tail‘𝐷)‘𝑣)))
6866, 67syl6ibr 241 . . . . . . . . . . . 12 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → (𝑥 ∈ ((tail‘𝐷)‘𝑤) → 𝑥 ∈ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
6968ssrdv 3574 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
70 sseq1 3589 . . . . . . . . . . . 12 (𝑧 = ((tail‘𝐷)‘𝑤) → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ ((tail‘𝐷)‘𝑤) ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣))))
7170rspcev 3282 . . . . . . . . . . 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 691 . . . . . . . . . 10 (((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) ∧ (𝑤𝑋 ∧ (𝑢𝐷𝑤𝑣𝐷𝑤))) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
7333, 72rexlimddv 3017 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)))
74 ineq1 3769 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑢) = 𝑥 → (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) = (𝑥 ∩ ((tail‘𝐷)‘𝑣)))
7574sseq2d 3596 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑢) = 𝑥 → (𝑧 ⊆ (((tail‘𝐷)‘𝑢) ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣))))
7675rexbidv 3034 . . . . . . . . . 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 3770 . . . . . . . . . . . 12 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑥 ∩ ((tail‘𝐷)‘𝑣)) = (𝑥𝑦))
7877sseq2d 3596 . . . . . . . . . . 11 (((tail‘𝐷)‘𝑣) = 𝑦 → (𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ 𝑧 ⊆ (𝑥𝑦)))
7978rexbidv 3034 . . . . . . . . . 10 (((tail‘𝐷)‘𝑣) = 𝑦 → (∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥 ∩ ((tail‘𝐷)‘𝑣)) ↔ ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8076, 79sylan9bb 732 . . . . . . . . 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 234 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ (𝑢𝑋𝑣𝑋)) → ((((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8281rexlimdvva 3020 . . . . . . 7 (𝐷 ∈ DirRel → (∃𝑢𝑋𝑣𝑋 (((tail‘𝐷)‘𝑢) = 𝑥 ∧ ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8331, 82syl5bir 232 . . . . . 6 (𝐷 ∈ DirRel → ((∃𝑢𝑋 ((tail‘𝐷)‘𝑢) = 𝑥 ∧ ∃𝑣𝑋 ((tail‘𝐷)‘𝑣) = 𝑦) → ∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦)))
8430, 83sylbid 229 . . . . 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 2953 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ∀𝑥 ∈ ran (tail‘𝐷)∀𝑦 ∈ ran (tail‘𝐷)∃𝑧 ∈ ran (tail‘𝐷)𝑧 ⊆ (𝑥𝑦))
8715, 26, 863jca 1235 . 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 6989 . . . . 5 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
891, 88syl5eqel 2692 . . . 4 (𝐷 ∈ DirRel → 𝑋 ∈ V)
9089adantr 480 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V)
91 isfbas2 21449 . . 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 959 1 ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  DirRelcdir 17051  tailctail 17052  fBascfbas 19555 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-dir 17053  df-tail 17054  df-fbas 19564 This theorem is referenced by:  filnetlem4  31546
 Copyright terms: Public domain W3C validator