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Theorem heibor1lem 35205
Description: Lemma for heibor1 35206. A compact metric space is complete. This proof works by considering the collection cls(𝐹 “ (ℤ𝑛)) for each 𝑛 ∈ ℕ, which has the finite intersection property because any finite intersection of upper integer sets is another upper integer set, so any finite intersection of the image closures will contain (𝐹 “ (ℤ𝑚)) for some 𝑚. Thus, by compactness, the intersection contains a point 𝑦, which must then be the convergent point of 𝐹. (Contributed by Jeff Madsen, 17-Jan-2014.) (Revised by Mario Carneiro, 5-Jun-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor1.3 (𝜑𝐷 ∈ (Met‘𝑋))
heibor1.4 (𝜑𝐽 ∈ Comp)
heibor1.5 (𝜑𝐹 ∈ (Cau‘𝐷))
heibor1.6 (𝜑𝐹:ℕ⟶𝑋)
Assertion
Ref Expression
heibor1lem (𝜑𝐹 ∈ dom (⇝𝑡𝐽))

Proof of Theorem heibor1lem
Dummy variables 𝑛 𝑦 𝑘 𝑟 𝑢 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor1.4 . . 3 (𝜑𝐽 ∈ Comp)
2 heibor1.3 . . . . . . . . . 10 (𝜑𝐷 ∈ (Met‘𝑋))
3 metxmet 22939 . . . . . . . . . 10 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
42, 3syl 17 . . . . . . . . 9 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 heibor.1 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
65mopntop 23045 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Top)
8 imassrn 5918 . . . . . . . . 9 (𝐹𝑢) ⊆ ran 𝐹
9 heibor1.6 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶𝑋)
109frnd 6501 . . . . . . . . . 10 (𝜑 → ran 𝐹𝑋)
115mopnuni 23046 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
124, 11syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1310, 12sseqtrd 3982 . . . . . . . . 9 (𝜑 → ran 𝐹 𝐽)
148, 13sstrid 3953 . . . . . . . 8 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
15 eqid 2822 . . . . . . . . 9 𝐽 = 𝐽
1615clscld 21650 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
177, 14, 16syl2anc 587 . . . . . . 7 (𝜑 → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
18 eleq1a 2909 . . . . . . 7 (((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽) → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
1917, 18syl 17 . . . . . 6 (𝜑 → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
2019rexlimdvw 3276 . . . . 5 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
2120abssdv 4020 . . . 4 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
22 fvex 6665 . . . . 5 (Clsd‘𝐽) ∈ V
2322elpw2 5224 . . . 4 ({𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽) ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
2421, 23sylibr 237 . . 3 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽))
25 elin 3924 . . . . . . 7 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin))
26 velpw 4516 . . . . . . . . 9 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ 𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
27 ssabral 4017 . . . . . . . . 9 (𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
2826, 27bitri 278 . . . . . . . 8 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
2928anbi1i 626 . . . . . . 7 ((𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin) ↔ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin))
3025, 29bitri 278 . . . . . 6 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin))
31 raleq 3386 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3231anbi2d 631 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
33 inteq 4854 . . . . . . . . . . . . . . 15 (𝑚 = ∅ → 𝑚 = ∅)
3433sseq2d 3974 . . . . . . . . . . . . . 14 (𝑚 = ∅ → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ ∅))
3534rexbidv 3283 . . . . . . . . . . . . 13 (𝑚 = ∅ → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅))
3632, 35imbi12d 348 . . . . . . . . . . . 12 (𝑚 = ∅ → (((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)))
37 raleq 3386 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3837anbi2d 631 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
39 inteq 4854 . . . . . . . . . . . . . . 15 (𝑚 = 𝑦 𝑚 = 𝑦)
4039sseq2d 3974 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑦))
4140rexbidv 3283 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦))
4238, 41imbi12d 348 . . . . . . . . . . . 12 (𝑚 = 𝑦 → (((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚) ↔ ((𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦)))
43 raleq 3386 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
4443anbi2d 631 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
45 inteq 4854 . . . . . . . . . . . . . . 15 (𝑚 = (𝑦 ∪ {𝑛}) → 𝑚 = (𝑦 ∪ {𝑛}))
4645sseq2d 3974 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4746rexbidv 3283 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4844, 47imbi12d 348 . . . . . . . . . . . 12 (𝑚 = (𝑦 ∪ {𝑛}) → (((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))))
49 raleq 3386 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
5049anbi2d 631 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
51 inteq 4854 . . . . . . . . . . . . . . 15 (𝑚 = 𝑟 𝑚 = 𝑟)
5251sseq2d 3974 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑟))
5352rexbidv 3283 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟))
5450, 53imbi12d 348 . . . . . . . . . . . 12 (𝑚 = 𝑟 → (((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚) ↔ ((𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟)))
55 uzf 12234 . . . . . . . . . . . . . . . 16 :ℤ⟶𝒫 ℤ
56 ffn 6494 . . . . . . . . . . . . . . . 16 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5755, 56ax-mp 5 . . . . . . . . . . . . . . 15 Fn ℤ
58 0z 11980 . . . . . . . . . . . . . . 15 0 ∈ ℤ
59 fnfvelrn 6830 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 0 ∈ ℤ) → (ℤ‘0) ∈ ran ℤ)
6057, 58, 59mp2an 691 . . . . . . . . . . . . . 14 (ℤ‘0) ∈ ran ℤ
61 ssv 3966 . . . . . . . . . . . . . . 15 (𝐹 “ (ℤ‘0)) ⊆ V
62 int0 4865 . . . . . . . . . . . . . . 15 ∅ = V
6361, 62sseqtrri 3979 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ‘0)) ⊆
64 imaeq2 5903 . . . . . . . . . . . . . . . 16 (𝑘 = (ℤ‘0) → (𝐹𝑘) = (𝐹 “ (ℤ‘0)))
6564sseq1d 3973 . . . . . . . . . . . . . . 15 (𝑘 = (ℤ‘0) → ((𝐹𝑘) ⊆ ∅ ↔ (𝐹 “ (ℤ‘0)) ⊆ ∅))
6665rspcev 3598 . . . . . . . . . . . . . 14 (((ℤ‘0) ∈ ran ℤ ∧ (𝐹 “ (ℤ‘0)) ⊆ ∅) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
6760, 63, 66mp2an 691 . . . . . . . . . . . . 13 𝑘 ∈ ran ℤ(𝐹𝑘) ⊆
6867a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
69 ssun1 4123 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑛})
70 ssralv 4008 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑛}) → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
7169, 70ax-mp 5 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
7271anim2i 619 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → (𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
7372imim1i 63 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦))
74 ssun2 4124 . . . . . . . . . . . . . . . . . 18 {𝑛} ⊆ (𝑦 ∪ {𝑛})
75 ssralv 4008 . . . . . . . . . . . . . . . . . 18 ({𝑛} ⊆ (𝑦 ∪ {𝑛}) → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
7674, 75ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
77 vex 3472 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
78 eqeq1 2826 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
7978rexbidv 3283 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
8077, 79ralsn 4593 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
8176, 80sylib 221 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
82 uzin2 14695 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ) → (𝑢𝑘) ∈ ran ℤ)
838, 10sstrid 3953 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐹𝑢) ⊆ 𝑋)
8483, 12sseqtrd 3982 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
8515sscls 21659 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
867, 84, 85syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
87 sseq2 3968 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → ((𝐹𝑢) ⊆ 𝑛 ↔ (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢))))
8886, 87syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝐹𝑢) ⊆ 𝑛))
89 inss2 4180 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑘
90 inss1 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑢
91 imass2 5943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑘 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘))
92 imass2 5943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑢 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢))
9391, 92anim12i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → ((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)))
94 ssin 4181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)) ↔ (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9593, 94sylib 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9689, 90, 95mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢))
97 ss2in 4187 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → ((𝐹𝑘) ∩ (𝐹𝑢)) ⊆ ( 𝑦𝑛))
9896, 97sstrid 3953 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ ( 𝑦𝑛))
9977intunsn 4890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∪ {𝑛}) = ( 𝑦𝑛)
10098, 99sseqtrrdi 3993 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛}))
101100expcom 417 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑢) ⊆ 𝑛 → ((𝐹𝑘) ⊆ 𝑦 → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
10288, 101syl6 35 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → ((𝐹𝑘) ⊆ 𝑦 → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛}))))
103102impd 414 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
104 imaeq2 5903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = (𝑢𝑘) → (𝐹𝑚) = (𝐹 “ (𝑢𝑘)))
105104sseq1d 3973 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑢𝑘) → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
106105rspcev 3598 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢𝑘) ∈ ran ℤ ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))
107106expcom 417 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛}) → ((𝑢𝑘) ∈ ran ℤ → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛})))
108103, 107syl6 35 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ((𝑢𝑘) ∈ ran ℤ → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))))
109108com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑢𝑘) ∈ ran ℤ → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))))
11082, 109syl5 34 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ) → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))))
111110rexlimdvv 3279 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ(𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛})))
112 reeanv 3348 . . . . . . . . . . . . . . . . . 18 (∃𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ(𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) ↔ (∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦))
113 imaeq2 5903 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
114113sseq1d 3973 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑘 → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
115114cbvrexvw 3425 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))
116111, 112, 1153imtr3g 298 . . . . . . . . . . . . . . . . 17 (𝜑 → ((∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
117116expd 419 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))))
11881, 117syl5 34 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))))
119118imp 410 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
12073, 119sylcom 30 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
121120a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ Fin → (((𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))))
12236, 42, 48, 54, 68, 121findcard2 8746 . . . . . . . . . . 11 (𝑟 ∈ Fin → ((𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟))
123122com12 32 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → (𝑟 ∈ Fin → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟))
124123impr 458 . . . . . . . . 9 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟)
1259ffnd 6495 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
126 inss1 4179 . . . . . . . . . . . . . . 15 (𝑘 ∩ ℕ) ⊆ 𝑘
127 imass2 5943 . . . . . . . . . . . . . . 15 ((𝑘 ∩ ℕ) ⊆ 𝑘 → (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘))
128126, 127ax-mp 5 . . . . . . . . . . . . . 14 (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘)
129 nnuz 12269 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
130 1z 12000 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
131 fnfvelrn 6830 . . . . . . . . . . . . . . . . . . . . 21 ((ℤ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ‘1) ∈ ran ℤ)
13257, 130, 131mp2an 691 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘1) ∈ ran ℤ
133129, 132eqeltri 2910 . . . . . . . . . . . . . . . . . . 19 ℕ ∈ ran ℤ
134 uzin2 14695 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ran ℤ ∧ ℕ ∈ ran ℤ) → (𝑘 ∩ ℕ) ∈ ran ℤ)
135133, 134mpan2 690 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ∈ ran ℤ)
136 uzn0 12248 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∩ ℕ) ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
137135, 136syl 17 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
138 n0 4282 . . . . . . . . . . . . . . . . 17 ((𝑘 ∩ ℕ) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
139137, 138sylib 221 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran ℤ → ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
140 fnfun 6432 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → Fun 𝐹)
141 inss2 4180 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∩ ℕ) ⊆ ℕ
142 fndm 6434 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn ℕ → dom 𝐹 = ℕ)
143141, 142sseqtrrid 3995 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → (𝑘 ∩ ℕ) ⊆ dom 𝐹)
144 funfvima2 6976 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (𝑘 ∩ ℕ) ⊆ dom 𝐹) → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
145140, 143, 144syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
146 ne0i 4272 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
147145, 146syl6 35 . . . . . . . . . . . . . . . . 17 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
148147exlimdv 1934 . . . . . . . . . . . . . . . 16 (𝐹 Fn ℕ → (∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
149139, 148syl5 34 . . . . . . . . . . . . . . 15 (𝐹 Fn ℕ → (𝑘 ∈ ran ℤ → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
150149imp 410 . . . . . . . . . . . . . 14 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
151 ssn0 4326 . . . . . . . . . . . . . 14 (((𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅) → (𝐹𝑘) ≠ ∅)
152128, 150, 151sylancr 590 . . . . . . . . . . . . 13 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹𝑘) ≠ ∅)
153 ssn0 4326 . . . . . . . . . . . . . 14 (((𝐹𝑘) ⊆ 𝑟 ∧ (𝐹𝑘) ≠ ∅) → 𝑟 ≠ ∅)
154153expcom 417 . . . . . . . . . . . . 13 ((𝐹𝑘) ≠ ∅ → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
155152, 154syl 17 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
156155rexlimdva 3270 . . . . . . . . . . 11 (𝐹 Fn ℕ → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
157125, 156syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
158157adantr 484 . . . . . . . . 9 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
159124, 158mpd 15 . . . . . . . 8 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → 𝑟 ≠ ∅)
160159necomd 3066 . . . . . . 7 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∅ ≠ 𝑟)
161160neneqd 3016 . . . . . 6 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ¬ ∅ = 𝑟)
16230, 161sylan2b 596 . . . . 5 ((𝜑𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)) → ¬ ∅ = 𝑟)
163162nrexdv 3256 . . . 4 (𝜑 → ¬ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
164 0ex 5187 . . . . 5 ∅ ∈ V
165 zex 11978 . . . . . . . 8 ℤ ∈ V
166165pwex 5258 . . . . . . 7 𝒫 ℤ ∈ V
167 frn 6500 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
16855, 167ax-mp 5 . . . . . . 7 ran ℤ ⊆ 𝒫 ℤ
169166, 168ssexi 5202 . . . . . 6 ran ℤ ∈ V
170169abrexex 7649 . . . . 5 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ V
171 elfi 8865 . . . . 5 ((∅ ∈ V ∧ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ V) → (∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟))
172164, 170, 171mp2an 691 . . . 4 (∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
173163, 172sylnibr 332 . . 3 (𝜑 → ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
174 cmptop 21998 . . . . . 6 (𝐽 ∈ Comp → 𝐽 ∈ Top)
175 cmpfi 22011 . . . . . 6 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅)))
176174, 175syl 17 . . . . 5 (𝐽 ∈ Comp → (𝐽 ∈ Comp ↔ ∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅)))
177176ibi 270 . . . 4 (𝐽 ∈ Comp → ∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅))
178 fveq2 6652 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (fi‘𝑚) = (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
179178eleq2d 2899 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (∅ ∈ (fi‘𝑚) ↔ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
180179notbid 321 . . . . . 6 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (¬ ∅ ∈ (fi‘𝑚) ↔ ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
181 inteq 4854 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → 𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
182181neeq1d 3070 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅))
183 n0 4282 . . . . . . 7 ( {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅ ↔ ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
184182, 183syl6bb 290 . . . . . 6 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
185180, 184imbi12d 348 . . . . 5 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ((¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
186185rspccv 3595 . . . 4 (∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅) → ({𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽) → (¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
187177, 186syl 17 . . 3 (𝐽 ∈ Comp → ({𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽) → (¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
1881, 24, 173, 187syl3c 66 . 2 (𝜑 → ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
189 lmrel 21833 . . 3 Rel (⇝𝑡𝐽)
190 r19.23v 3265 . . . . . 6 (∀𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
191190albii 1821 . . . . 5 (∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
192 fvex 6665 . . . . . . . 8 ((cls‘𝐽)‘(𝐹𝑢)) ∈ V
193 eleq2 2902 . . . . . . . 8 (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝑦𝑘𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))))
194192, 193ceqsalv 3507 . . . . . . 7 (∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
195194ralbii 3157 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
196 ralcom4 3223 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
197195, 196bitr3i 280 . . . . 5 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
198 vex 3472 . . . . . 6 𝑦 ∈ V
199198elintab 4862 . . . . 5 (𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
200191, 197, 1993bitr4i 306 . . . 4 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
201 eqid 2822 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))
202 imaeq2 5903 . . . . . . . . . . . . 13 (𝑢 = ℕ → (𝐹𝑢) = (𝐹 “ ℕ))
203202fveq2d 6656 . . . . . . . . . . . 12 (𝑢 = ℕ → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ ℕ)))
204203rspceeqv 3613 . . . . . . . . . . 11 ((ℕ ∈ ran ℤ ∧ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))) → ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢)))
205133, 201, 204mp2an 691 . . . . . . . . . 10 𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))
206 fvex 6665 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ V
207 eqeq1 2826 . . . . . . . . . . . 12 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
208207rexbidv 3283 . . . . . . . . . . 11 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
209206, 208elab 3642 . . . . . . . . . 10 (((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢)))
210205, 209mpbir 234 . . . . . . . . 9 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}
211 intss1 4866 . . . . . . . . 9 (((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ ((cls‘𝐽)‘(𝐹 “ ℕ)))
212210, 211ax-mp 5 . . . . . . . 8 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ ((cls‘𝐽)‘(𝐹 “ ℕ))
213 imassrn 5918 . . . . . . . . . . 11 (𝐹 “ ℕ) ⊆ ran 𝐹
214213, 13sstrid 3953 . . . . . . . . . 10 (𝜑 → (𝐹 “ ℕ) ⊆ 𝐽)
21515clsss3 21662 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹 “ ℕ) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
2167, 214, 215syl2anc 587 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
217216, 12sseqtrrd 3983 . . . . . . . 8 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝑋)
218212, 217sstrid 3953 . . . . . . 7 (𝜑 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ 𝑋)
219218sselda 3942 . . . . . 6 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝑦𝑋)
220200, 219sylan2b 596 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝑦𝑋)
221 heibor1.5 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (Cau‘𝐷))
222 1zzd 12001 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
223129, 4, 222iscau3 23880 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦))))
224221, 223mpbid 235 . . . . . . . . . . 11 (𝜑 → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)))
225224simprd 499 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦))
226 simp3 1135 . . . . . . . . . . . . 13 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
227226ralimi 3152 . . . . . . . . . . . 12 (∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
228227reximi 3231 . . . . . . . . . . 11 (∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
229228ralimi 3152 . . . . . . . . . 10 (∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
230225, 229syl 17 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
231230adantr 484 . . . . . . . 8 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
232 rphalfcl 12404 . . . . . . . 8 (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
233 breq2 5046 . . . . . . . . . . 11 (𝑦 = (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
2342332ralbidv 3189 . . . . . . . . . 10 (𝑦 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
235234rexbidv 3283 . . . . . . . . 9 (𝑦 = (𝑟 / 2) → (∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
236235rspccva 3597 . . . . . . . 8 ((∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ∧ (𝑟 / 2) ∈ ℝ+) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2))
237231, 232, 236syl2an 598 . . . . . . 7 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2))
2389ffund 6498 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
239238ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → Fun 𝐹)
2407ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐽 ∈ Top)
241 imassrn 5918 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ𝑚)) ⊆ ran 𝐹
242241, 13sstrid 3953 . . . . . . . . . . . . 13 (𝜑 → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
243242ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
244 nnz 11992 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
245 fnfvelrn 6830 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 𝑚 ∈ ℤ) → (ℤ𝑚) ∈ ran ℤ)
24657, 244, 245sylancr 590 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ∈ ran ℤ)
247246ad2antll 728 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ∈ ran ℤ)
248 simplr 768 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
249 imaeq2 5903 . . . . . . . . . . . . . . . 16 (𝑢 = (ℤ𝑚) → (𝐹𝑢) = (𝐹 “ (ℤ𝑚)))
250249fveq2d 6656 . . . . . . . . . . . . . . 15 (𝑢 = (ℤ𝑚) → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚))))
251250eleq2d 2899 . . . . . . . . . . . . . 14 (𝑢 = (ℤ𝑚) → (𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))))
252251rspcv 3593 . . . . . . . . . . . . 13 ((ℤ𝑚) ∈ ran ℤ → (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))))
253247, 248, 252sylc 65 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚))))
2544ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐷 ∈ (∞Met‘𝑋))
255220adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦𝑋)
256232ad2antrl 727 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ+)
257256rpxrd 12420 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ*)
2585blopn 23105 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ*) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
259254, 255, 257, 258syl3anc 1368 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
260 blcntr 23018 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
261254, 255, 256, 260syl3anc 1368 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
26215clsndisj 21678 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝐹 “ (ℤ𝑚)) ⊆ 𝐽𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))) ∧ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅)
263240, 243, 253, 259, 261, 262syl32anc 1375 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅)
264 n0 4282 . . . . . . . . . . . 12 (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))))
265 inss2 4180 . . . . . . . . . . . . . . . . 17 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝐹 “ (ℤ𝑚))
266265sseli 3938 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝐹 “ (ℤ𝑚)))
267 fvelima 6713 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑛 ∈ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
268266, 267sylan2 595 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
269 inss1 4179 . . . . . . . . . . . . . . . . . . 19 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝑦(ball‘𝐷)(𝑟 / 2))
270269sseli 3938 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
271270adantl 485 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
272 eleq1a 2909 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → ((𝐹𝑘) = 𝑛 → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
273271, 272syl 17 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → ((𝐹𝑘) = 𝑛 → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
274273reximdv 3259 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → (∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛 → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
275268, 274mpd 15 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
276275ex 416 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
277276exlimdv 1934 . . . . . . . . . . . 12 (Fun 𝐹 → (∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
278264, 277syl5bi 245 . . . . . . . . . . 11 (Fun 𝐹 → (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅ → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
279239, 263, 278sylc 65 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
280 r19.29 3242 . . . . . . . . . . 11 ((∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ (ℤ𝑚)(∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
281 uznnssnn 12283 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ⊆ ℕ)
282281ad2antll 728 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ⊆ ℕ)
283 simprlr 779 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
2844ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
285 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℝ+)
286285, 232syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ+)
287286rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ*)
288 simpllr 775 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑦𝑋)
2899ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐹:ℕ⟶𝑋)
290 eluznn 12306 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑚)) → 𝑘 ∈ ℕ)
291290ad2ant2lr 747 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑟 ∈ ℝ+𝑚 ∈ ℕ) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → 𝑘 ∈ ℕ)
292291ad2ant2lr 747 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑘 ∈ ℕ)
293289, 292ffvelrnd 6834 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ 𝑋)
294 elbl3 22997 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑟 / 2) ∈ ℝ*) ∧ (𝑦𝑋 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) ↔ ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2)))
295284, 287, 288, 293, 294syl22anc 837 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) ↔ ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2)))
296283, 295mpbid 235 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2))
2972ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (Met‘𝑋))
298 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ (ℤ𝑘))
299 eluznn 12306 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (ℤ𝑘)) → 𝑛 ∈ ℕ)
300292, 298, 299syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ ℕ)
301289, 300ffvelrnd 6834 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑛) ∈ 𝑋)
302 metcl 22937 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
303297, 293, 301, 302syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
304 metcl 22937 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋𝑦𝑋) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
305297, 293, 288, 304syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
306286rpred 12419 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ)
307 lt2add 11114 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ ∧ ((𝐹𝑘)𝐷𝑦) ∈ ℝ) ∧ ((𝑟 / 2) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ)) → ((((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2)) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2))))
308303, 305, 306, 306, 307syl22anc 837 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2)) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2))))
309296, 308mpan2d 693 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2))))
310285rpcnd 12421 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℂ)
3113102halvesd 11871 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝑟 / 2) + (𝑟 / 2)) = 𝑟)
312311breq2d 5054 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2)) ↔ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < 𝑟))
313309, 312sylibd 242 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < 𝑟))
314 mettri2 22946 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋)) → ((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)))
315297, 293, 301, 288, 314syl13anc 1369 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)))
316 metcl 22937 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
317297, 301, 288, 316syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
318303, 305readdcld 10659 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∈ ℝ)
319285rpred 12419 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℝ)
320 lelttr 10720 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹𝑛)𝐷𝑦) ∈ ℝ ∧ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∧ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < 𝑟) → ((𝐹𝑛)𝐷𝑦) < 𝑟))
321317, 318, 319, 320syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∧ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < 𝑟) → ((𝐹𝑛)𝐷𝑦) < 𝑟))
322315, 321mpand 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) < 𝑟 → ((𝐹𝑛)𝐷𝑦) < 𝑟))
323313, 322syld 47 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ((𝐹𝑛)𝐷𝑦) < 𝑟))
324323anassrs 471 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) ∧ 𝑛 ∈ (ℤ𝑘)) → (((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ((𝐹𝑛)𝐷𝑦) < 𝑟))
325324ralimdva 3169 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → (∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
326325expr 460 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ 𝑘 ∈ (ℤ𝑚)) → ((𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → (∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
327326com23 86 . . . . . . . . . . . . . . 15 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ 𝑘 ∈ (ℤ𝑚)) → (∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ((𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
328327impd 414 . . . . . . . . . . . . . 14 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ 𝑘 ∈ (ℤ𝑚)) → ((∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
329328reximdva 3260 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (∃𝑘 ∈ (ℤ𝑚)(∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
330 ssrexv 4009 . . . . . . . . . . . . 13 ((ℤ𝑚) ⊆ ℕ → (∃𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟 → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
331282, 329, 330sylsyld 61 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (∃𝑘 ∈ (ℤ𝑚)(∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
332220, 331syldanl 604 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (∃𝑘 ∈ (ℤ𝑚)(∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
333280, 332syl5 34 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ((∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) ∧ ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
334279, 333mpan2d 693 . . . . . . . . 9 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
335334anassrs 471 . . . . . . . 8 ((((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
336335rexlimdva 3270 . . . . . . 7 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) → (∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟))
337237, 336mpd 15 . . . . . 6 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)
338337ralrimiva 3174 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)
339 eqidd 2823 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = (𝐹𝑛))
3405, 4, 129, 222, 339, 9lmmbrf 23864 . . . . . 6 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
341340adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
342220, 338, 341mpbir2and 712 . . . 4 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝐹(⇝𝑡𝐽)𝑦)
343200, 342sylan2br 597 . . 3 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹(⇝𝑡𝐽)𝑦)
344 releldm 5791 . . 3 ((Rel (⇝𝑡𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ dom (⇝𝑡𝐽))
345189, 343, 344sylancr 590 . 2 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹 ∈ dom (⇝𝑡𝐽))
346188, 345exlimddv 1936 1 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2114  {cab 2800  wne 3011  wral 3130  wrex 3131  Vcvv 3469  cun 3906  cin 3907  wss 3908  c0 4265  𝒫 cpw 4511  {csn 4539   cuni 4813   cint 4851   class class class wbr 5042  dom cdm 5532  ran crn 5533  cima 5535  Rel wrel 5537  Fun wfun 6328   Fn wfn 6329  wf 6330  cfv 6334  (class class class)co 7140  pm cpm 8394  Fincfn 8496  ficfi 8862  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cle 10665   / cdiv 11286  cn 11625  2c2 11680  cz 11969  cuz 12231  +crp 12377  ∞Metcxmet 20074  Metcmet 20075  ballcbl 20076  MetOpencmopn 20079  Topctop 21496  Clsdccld 21619  clsccl 21621  𝑡clm 21829  Compccmp 21989  Cauccau 23855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-top 21497  df-topon 21514  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-lm 21832  df-cmp 21990  df-cau 23858
This theorem is referenced by:  heibor1  35206
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