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Theorem heibor1lem 35995
Description: Lemma for heibor1 35996. 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 23515 . . . . . . . . . 10 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
42, 3syl 17 . . . . . . . . 9 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 heibor.1 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
65mopntop 23621 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Top)
8 imassrn 5981 . . . . . . . . 9 (𝐹𝑢) ⊆ ran 𝐹
9 heibor1.6 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶𝑋)
109frnd 6626 . . . . . . . . . 10 (𝜑 → ran 𝐹𝑋)
115mopnuni 23622 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
124, 11syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1310, 12sseqtrd 3963 . . . . . . . . 9 (𝜑 → ran 𝐹 𝐽)
148, 13sstrid 3934 . . . . . . . 8 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
15 eqid 2733 . . . . . . . . 9 𝐽 = 𝐽
1615clscld 22226 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
177, 14, 16syl2anc 583 . . . . . . 7 (𝜑 → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
18 eleq1a 2829 . . . . . . 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 3151 . . . . 5 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
2120abssdv 4004 . . . 4 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
22 fvex 6805 . . . . 5 (Clsd‘𝐽) ∈ V
2322elpw2 5272 . . . 4 ({𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽) ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
2421, 23sylibr 233 . . 3 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽))
25 elin 3905 . . . . . . 7 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin))
26 velpw 4541 . . . . . . . . 9 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ 𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
27 ssabral 3998 . . . . . . . . 9 (𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
2826, 27bitri 274 . . . . . . . 8 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
2928anbi1i 623 . . . . . . 7 ((𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin) ↔ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin))
3025, 29bitri 274 . . . . . 6 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin))
31 raleq 3344 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3231anbi2d 628 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
33 inteq 4885 . . . . . . . . . . . . . . 15 (𝑚 = ∅ → 𝑚 = ∅)
3433sseq2d 3955 . . . . . . . . . . . . . 14 (𝑚 = ∅ → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ ∅))
3534rexbidv 3169 . . . . . . . . . . . . 13 (𝑚 = ∅ → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅))
3632, 35imbi12d 344 . . . . . . . . . . . 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 3344 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3837anbi2d 628 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
39 inteq 4885 . . . . . . . . . . . . . . 15 (𝑚 = 𝑦 𝑚 = 𝑦)
4039sseq2d 3955 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑦))
4140rexbidv 3169 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦))
4238, 41imbi12d 344 . . . . . . . . . . . 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 3344 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
4443anbi2d 628 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
45 inteq 4885 . . . . . . . . . . . . . . 15 (𝑚 = (𝑦 ∪ {𝑛}) → 𝑚 = (𝑦 ∪ {𝑛}))
4645sseq2d 3955 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4746rexbidv 3169 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4844, 47imbi12d 344 . . . . . . . . . . . 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 3344 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
5049anbi2d 628 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
51 inteq 4885 . . . . . . . . . . . . . . 15 (𝑚 = 𝑟 𝑚 = 𝑟)
5251sseq2d 3955 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑟))
5352rexbidv 3169 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑚 ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟))
5450, 53imbi12d 344 . . . . . . . . . . . 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 12613 . . . . . . . . . . . . . . . 16 :ℤ⟶𝒫 ℤ
56 ffn 6618 . . . . . . . . . . . . . . . 16 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5755, 56ax-mp 5 . . . . . . . . . . . . . . 15 Fn ℤ
58 0z 12358 . . . . . . . . . . . . . . 15 0 ∈ ℤ
59 fnfvelrn 6978 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 0 ∈ ℤ) → (ℤ‘0) ∈ ran ℤ)
6057, 58, 59mp2an 688 . . . . . . . . . . . . . 14 (ℤ‘0) ∈ ran ℤ
61 ssv 3947 . . . . . . . . . . . . . . 15 (𝐹 “ (ℤ‘0)) ⊆ V
62 int0 4896 . . . . . . . . . . . . . . 15 ∅ = V
6361, 62sseqtrri 3960 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ‘0)) ⊆
64 imaeq2 5966 . . . . . . . . . . . . . . . 16 (𝑘 = (ℤ‘0) → (𝐹𝑘) = (𝐹 “ (ℤ‘0)))
6564sseq1d 3954 . . . . . . . . . . . . . . 15 (𝑘 = (ℤ‘0) → ((𝐹𝑘) ⊆ ∅ ↔ (𝐹 “ (ℤ‘0)) ⊆ ∅))
6665rspcev 3563 . . . . . . . . . . . . . 14 (((ℤ‘0) ∈ ran ℤ ∧ (𝐹 “ (ℤ‘0)) ⊆ ∅) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
6760, 63, 66mp2an 688 . . . . . . . . . . . . 13 𝑘 ∈ ran ℤ(𝐹𝑘) ⊆
6867a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
69 ssun1 4109 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑛})
70 ssralv 3989 . . . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . 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 4110 . . . . . . . . . . . . . . . . . 18 {𝑛} ⊆ (𝑦 ∪ {𝑛})
75 ssralv 3989 . . . . . . . . . . . . . . . . . 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 3438 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
78 eqeq1 2737 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
7978rexbidv 3169 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
8077, 79ralsn 4620 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
8176, 80sylib 217 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
82 uzin2 15084 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ) → (𝑢𝑘) ∈ ran ℤ)
838, 10sstrid 3934 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐹𝑢) ⊆ 𝑋)
8483, 12sseqtrd 3963 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
8515sscls 22235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
867, 84, 85syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
87 sseq2 3949 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → ((𝐹𝑢) ⊆ 𝑛 ↔ (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢))))
8886, 87syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝐹𝑢) ⊆ 𝑛))
89 inss2 4166 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑘
90 inss1 4165 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑢
91 imass2 6011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑘 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘))
92 imass2 6011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑢 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢))
9391, 92anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → ((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)))
94 ssin 4167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)) ↔ (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9593, 94sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9689, 90, 95mp2an 688 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢))
97 ss2in 4173 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → ((𝐹𝑘) ∩ (𝐹𝑢)) ⊆ ( 𝑦𝑛))
9896, 97sstrid 3934 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ ( 𝑦𝑛))
9977intunsn 4923 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∪ {𝑛}) = ( 𝑦𝑛)
10098, 99sseqtrrdi 3974 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛}))
101100expcom 413 . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
104 imaeq2 5966 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = (𝑢𝑘) → (𝐹𝑚) = (𝐹 “ (𝑢𝑘)))
105104sseq1d 3954 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑢𝑘) → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
106105rspcev 3563 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢𝑘) ∈ ran ℤ ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))
107106expcom 413 . . . . . . . . . . . . . . . . . . . . . 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 3198 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ(𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛})))
112 reeanv 3211 . . . . . . . . . . . . . . . . . 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 5966 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
114113sseq1d 3954 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑘 → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
115114cbvrexvw 3386 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))
116111, 112, 1153imtr3g 294 . . . . . . . . . . . . . . . . 17 (𝜑 → ((∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑦) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
117116expd 415 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 8972 . . . . . . . . . . 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 454 . . . . . . . . 9 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟)
1259ffnd 6619 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
126 inss1 4165 . . . . . . . . . . . . . . 15 (𝑘 ∩ ℕ) ⊆ 𝑘
127 imass2 6011 . . . . . . . . . . . . . . 15 ((𝑘 ∩ ℕ) ⊆ 𝑘 → (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘))
128126, 127ax-mp 5 . . . . . . . . . . . . . 14 (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘)
129 nnuz 12649 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
130 1z 12378 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
131 fnfvelrn 6978 . . . . . . . . . . . . . . . . . . . . 21 ((ℤ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ‘1) ∈ ran ℤ)
13257, 130, 131mp2an 688 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘1) ∈ ran ℤ
133129, 132eqeltri 2830 . . . . . . . . . . . . . . . . . . 19 ℕ ∈ ran ℤ
134 uzin2 15084 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ran ℤ ∧ ℕ ∈ ran ℤ) → (𝑘 ∩ ℕ) ∈ ran ℤ)
135133, 134mpan2 687 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ∈ ran ℤ)
136 uzn0 12627 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∩ ℕ) ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
137135, 136syl 17 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
138 n0 4283 . . . . . . . . . . . . . . . . 17 ((𝑘 ∩ ℕ) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
139137, 138sylib 217 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran ℤ → ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
140 fnfun 6552 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → Fun 𝐹)
141 inss2 4166 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∩ ℕ) ⊆ ℕ
142 fndm 6555 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn ℕ → dom 𝐹 = ℕ)
143141, 142sseqtrrid 3976 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → (𝑘 ∩ ℕ) ⊆ dom 𝐹)
144 funfvima2 7127 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (𝑘 ∩ ℕ) ⊆ dom 𝐹) → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
145140, 143, 144syl2anc 583 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
146 ne0i 4271 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
147145, 146syl6 35 . . . . . . . . . . . . . . . . 17 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
148147exlimdv 1932 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 14 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
151 ssn0 4337 . . . . . . . . . . . . . 14 (((𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅) → (𝐹𝑘) ≠ ∅)
152128, 150, 151sylancr 586 . . . . . . . . . . . . 13 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹𝑘) ≠ ∅)
153 ssn0 4337 . . . . . . . . . . . . . 14 (((𝐹𝑘) ⊆ 𝑟 ∧ (𝐹𝑘) ≠ ∅) → 𝑟 ≠ ∅)
154153expcom 413 . . . . . . . . . . . . 13 ((𝐹𝑘) ≠ ∅ → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
155152, 154syl 17 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
156155rexlimdva 3146 . . . . . . . . . . 11 (𝐹 Fn ℕ → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
157125, 156syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
158157adantr 480 . . . . . . . . 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 2994 . . . . . . 7 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∅ ≠ 𝑟)
161160neneqd 2943 . . . . . 6 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ¬ ∅ = 𝑟)
16230, 161sylan2b 593 . . . . 5 ((𝜑𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)) → ¬ ∅ = 𝑟)
163162nrexdv 3140 . . . 4 (𝜑 → ¬ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
164 0ex 5234 . . . . 5 ∅ ∈ V
165 zex 12356 . . . . . . . 8 ℤ ∈ V
166165pwex 5306 . . . . . . 7 𝒫 ℤ ∈ V
167 frn 6625 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
16855, 167ax-mp 5 . . . . . . 7 ran ℤ ⊆ 𝒫 ℤ
169166, 168ssexi 5249 . . . . . 6 ran ℤ ∈ V
170169abrexex 7825 . . . . 5 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ V
171 elfi 9200 . . . . 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 688 . . . 4 (∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
173163, 172sylnibr 328 . . 3 (𝜑 → ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
174 cmptop 22574 . . . . . 6 (𝐽 ∈ Comp → 𝐽 ∈ Top)
175 cmpfi 22587 . . . . . 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 266 . . . 4 (𝐽 ∈ Comp → ∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅))
178 fveq2 6792 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (fi‘𝑚) = (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
179178eleq2d 2819 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (∅ ∈ (fi‘𝑚) ↔ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
180179notbid 317 . . . . . 6 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (¬ ∅ ∈ (fi‘𝑚) ↔ ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
181 inteq 4885 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → 𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
182181neeq1d 2998 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅))
183 n0 4283 . . . . . . 7 ( {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅ ↔ ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
184182, 183bitrdi 286 . . . . . 6 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
185180, 184imbi12d 344 . . . . 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 3560 . . . 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 22409 . . 3 Rel (⇝𝑡𝐽)
190 r19.23v 3173 . . . . . 6 (∀𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
191190albii 1817 . . . . 5 (∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
192 fvex 6805 . . . . . . . 8 ((cls‘𝐽)‘(𝐹𝑢)) ∈ V
193 eleq2 2822 . . . . . . . 8 (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝑦𝑘𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))))
194192, 193ceqsalv 3469 . . . . . . 7 (∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
195194ralbii 3090 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
196 ralcom4 3258 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
197195, 196bitr3i 276 . . . . 5 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
198 vex 3438 . . . . . 6 𝑦 ∈ V
199198elintab 4893 . . . . 5 (𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
200191, 197, 1993bitr4i 302 . . . 4 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
201 eqid 2733 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))
202 imaeq2 5966 . . . . . . . . . . . . 13 (𝑢 = ℕ → (𝐹𝑢) = (𝐹 “ ℕ))
203202fveq2d 6796 . . . . . . . . . . . 12 (𝑢 = ℕ → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ ℕ)))
204203rspceeqv 3577 . . . . . . . . . . 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 688 . . . . . . . . . 10 𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))
206 fvex 6805 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ V
207 eqeq1 2737 . . . . . . . . . . . 12 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
208207rexbidv 3169 . . . . . . . . . . 11 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
209206, 208elab 3611 . . . . . . . . . 10 (((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢)))
210205, 209mpbir 230 . . . . . . . . 9 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}
211 intss1 4897 . . . . . . . . 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 5981 . . . . . . . . . . 11 (𝐹 “ ℕ) ⊆ ran 𝐹
214213, 13sstrid 3934 . . . . . . . . . 10 (𝜑 → (𝐹 “ ℕ) ⊆ 𝐽)
21515clsss3 22238 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹 “ ℕ) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
2167, 214, 215syl2anc 583 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
217216, 12sseqtrrd 3964 . . . . . . . 8 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝑋)
218212, 217sstrid 3934 . . . . . . 7 (𝜑 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ 𝑋)
219218sselda 3923 . . . . . 6 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝑦𝑋)
220200, 219sylan2b 593 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝑦𝑋)
221 heibor1.5 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (Cau‘𝐷))
222 1zzd 12379 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
223129, 4, 222iscau3 24470 . . . . . . . . . . . 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 231 . . . . . . . . . . 11 (𝜑 → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)))
225224simprd 495 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦))
226 simp3 1136 . . . . . . . . . . . . 13 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
227226ralimi 3080 . . . . . . . . . . . 12 (∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
228227reximi 3081 . . . . . . . . . . 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 3080 . . . . . . . . . 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 480 . . . . . . . 8 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
232 rphalfcl 12785 . . . . . . . 8 (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
233 breq2 5081 . . . . . . . . . . 11 (𝑦 = (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
2342332ralbidv 3206 . . . . . . . . . 10 (𝑦 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
235234rexbidv 3169 . . . . . . . . 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 3562 . . . . . . . 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 595 . . . . . . 7 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2))
2389ffund 6622 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
239238ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → Fun 𝐹)
2407ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐽 ∈ Top)
241 imassrn 5981 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ𝑚)) ⊆ ran 𝐹
242241, 13sstrid 3934 . . . . . . . . . . . . 13 (𝜑 → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
243242ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
244 nnz 12370 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
245 fnfvelrn 6978 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 𝑚 ∈ ℤ) → (ℤ𝑚) ∈ ran ℤ)
24657, 244, 245sylancr 586 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ∈ ran ℤ)
247246ad2antll 725 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ∈ ran ℤ)
248 simplr 765 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
249 imaeq2 5966 . . . . . . . . . . . . . . . 16 (𝑢 = (ℤ𝑚) → (𝐹𝑢) = (𝐹 “ (ℤ𝑚)))
250249fveq2d 6796 . . . . . . . . . . . . . . 15 (𝑢 = (ℤ𝑚) → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚))))
251250eleq2d 2819 . . . . . . . . . . . . . 14 (𝑢 = (ℤ𝑚) → (𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))))
252251rspcv 3559 . . . . . . . . . . . . 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 722 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐷 ∈ (∞Met‘𝑋))
255220adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦𝑋)
256232ad2antrl 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ+)
257256rpxrd 12801 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ*)
2585blopn 23684 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ*) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
259254, 255, 257, 258syl3anc 1369 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
260 blcntr 23594 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
261254, 255, 256, 260syl3anc 1369 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
26215clsndisj 22254 . . . . . . . . . . . 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 1376 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅)
264 n0 4283 . . . . . . . . . . . 12 (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))))
265 inss2 4166 . . . . . . . . . . . . . . . . 17 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝐹 “ (ℤ𝑚))
266265sseli 3919 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝐹 “ (ℤ𝑚)))
267 fvelima 6855 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑛 ∈ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
268266, 267sylan2 592 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
269 inss1 4165 . . . . . . . . . . . . . . . . . . 19 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝑦(ball‘𝐷)(𝑟 / 2))
270269sseli 3919 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
271270adantl 481 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
272 eleq1a 2829 . . . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
277276exlimdv 1932 . . . . . . . . . . . 12 (Fun 𝐹 → (∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
278264, 277syl5bi 241 . . . . . . . . . . 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 3111 . . . . . . . . . . 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 12663 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ⊆ ℕ)
282281ad2antll 725 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ⊆ ℕ)
283 simprlr 776 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
2844ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
285 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . . . 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 12801 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ*)
288 simpllr 772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑦𝑋)
2899ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐹:ℕ⟶𝑋)
290 eluznn 12686 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑚)) → 𝑘 ∈ ℕ)
291290ad2ant2lr 744 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑟 ∈ ℝ+𝑚 ∈ ℕ) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → 𝑘 ∈ ℕ)
292291ad2ant2lr 744 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑘 ∈ ℕ)
293289, 292ffvelcdmd 6982 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ 𝑋)
294 elbl3 23573 . . . . . . . . . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2))
2972ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (Met‘𝑋))
298 simprr 769 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ (ℤ𝑘))
299 eluznn 12686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (ℤ𝑘)) → 𝑛 ∈ ℕ)
300292, 298, 299syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ ℕ)
301289, 300ffvelcdmd 6982 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑛) ∈ 𝑋)
302 metcl 23513 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
303297, 293, 301, 302syl3anc 1369 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
304 metcl 23513 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋𝑦𝑋) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
305297, 293, 288, 304syl3anc 1369 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
306286rpred 12800 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ)
307 lt2add 11488 . . . . . . . . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . 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 12802 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℂ)
3113102halvesd 12247 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝑟 / 2) + (𝑟 / 2)) = 𝑟)
312311breq2d 5089 . . . . . . . . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . . . . . . . . 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 23522 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋)) → ((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)))
315297, 293, 301, 288, 314syl13anc 1370 . . . . . . . . . . . . . . . . . . . . 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 23513 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
317297, 301, 288, 316syl3anc 1369 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
318303, 305readdcld 11032 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∈ ℝ)
319285rpred 12800 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℝ)
320 lelttr 11093 . . . . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . 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 3158 . . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . 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 3159 . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . 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 601 . . . . . . . . . . 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 690 . . . . . . . . 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 467 . . . . . . . 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 3146 . . . . . . 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 3137 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)
339 eqidd 2734 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = (𝐹𝑛))
3405, 4, 129, 222, 339, 9lmmbrf 24454 . . . . . 6 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
341340adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
342220, 338, 341mpbir2and 709 . . . 4 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝐹(⇝𝑡𝐽)𝑦)
343200, 342sylan2br 594 . . 3 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹(⇝𝑡𝐽)𝑦)
344 releldm 5856 . . 3 ((Rel (⇝𝑡𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ dom (⇝𝑡𝐽))
345189, 343, 344sylancr 586 . 2 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹 ∈ dom (⇝𝑡𝐽))
346188, 345exlimddv 1934 1 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085  wal 1535   = wceq 1537  wex 1777  wcel 2101  {cab 2710  wne 2938  wral 3059  wrex 3068  Vcvv 3434  cun 3887  cin 3888  wss 3889  c0 4259  𝒫 cpw 4536  {csn 4564   cuni 4841   cint 4882   class class class wbr 5077  dom cdm 5591  ran crn 5592  cima 5594  Rel wrel 5596  Fun wfun 6441   Fn wfn 6442  wf 6443  cfv 6447  (class class class)co 7295  pm cpm 8636  Fincfn 8753  ficfi 9197  cc 10897  cr 10898  0cc0 10899  1c1 10900   + caddc 10902  *cxr 11036   < clt 11037  cle 11038   / cdiv 11660  cn 12001  2c2 12056  cz 12347  cuz 12610  +crp 12758  ∞Metcxmet 20610  Metcmet 20611  ballcbl 20612  MetOpencmopn 20615  Topctop 22070  Clsdccld 22195  clsccl 22197  𝑡clm 22405  Compccmp 22565  Cauccau 24445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-cnex 10955  ax-resscn 10956  ax-1cn 10957  ax-icn 10958  ax-addcl 10959  ax-addrcl 10960  ax-mulcl 10961  ax-mulrcl 10962  ax-mulcom 10963  ax-addass 10964  ax-mulass 10965  ax-distr 10966  ax-i2m1 10967  ax-1ne0 10968  ax-1rid 10969  ax-rnegex 10970  ax-rrecex 10971  ax-cnre 10972  ax-pre-lttri 10973  ax-pre-lttrn 10974  ax-pre-ltadd 10975  ax-pre-mulgt0 10976  ax-pre-sup 10977
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3222  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-int 4883  df-iun 4929  df-iin 4930  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-om 7733  df-1st 7851  df-2nd 7852  df-frecs 8117  df-wrecs 8148  df-recs 8222  df-rdg 8261  df-1o 8317  df-er 8518  df-map 8637  df-pm 8638  df-en 8754  df-dom 8755  df-sdom 8756  df-fin 8757  df-fi 9198  df-sup 9229  df-inf 9230  df-pnf 11039  df-mnf 11040  df-xr 11041  df-ltxr 11042  df-le 11043  df-sub 11235  df-neg 11236  df-div 11661  df-nn 12002  df-2 12064  df-n0 12262  df-z 12348  df-uz 12611  df-q 12717  df-rp 12759  df-xneg 12876  df-xadd 12877  df-xmul 12878  df-topgen 17182  df-psmet 20617  df-xmet 20618  df-met 20619  df-bl 20620  df-mopn 20621  df-top 22071  df-topon 22088  df-bases 22124  df-cld 22198  df-ntr 22199  df-cls 22200  df-lm 22408  df-cmp 22566  df-cau 24448
This theorem is referenced by:  heibor1  35996
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