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Theorem heibor1lem 36268
Description: Lemma for heibor1 36269. 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 23687 . . . . . . . . . 10 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
42, 3syl 17 . . . . . . . . 9 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 heibor.1 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
65mopntop 23793 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Top)
8 imassrn 6024 . . . . . . . . 9 (𝐹𝑢) ⊆ ran 𝐹
9 heibor1.6 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶𝑋)
109frnd 6676 . . . . . . . . . 10 (𝜑 → ran 𝐹𝑋)
115mopnuni 23794 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
124, 11syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1310, 12sseqtrd 3984 . . . . . . . . 9 (𝜑 → ran 𝐹 𝐽)
148, 13sstrid 3955 . . . . . . . 8 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
15 eqid 2736 . . . . . . . . 9 𝐽 = 𝐽
1615clscld 22398 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
177, 14, 16syl2anc 584 . . . . . . 7 (𝜑 → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
18 eleq1a 2833 . . . . . . 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 3157 . . . . 5 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
2120abssdv 4025 . . . 4 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
22 fvex 6855 . . . . 5 (Clsd‘𝐽) ∈ V
2322elpw2 5302 . . . 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 3926 . . . . . . 7 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin))
26 velpw 4565 . . . . . . . . 9 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ 𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
27 ssabral 4019 . . . . . . . . 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 624 . . . . . . 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 3309 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3231anbi2d 629 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
33 inteq 4910 . . . . . . . . . . . . . . 15 (𝑚 = ∅ → 𝑚 = ∅)
3433sseq2d 3976 . . . . . . . . . . . . . 14 (𝑚 = ∅ → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ ∅))
3534rexbidv 3175 . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3837anbi2d 629 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
39 inteq 4910 . . . . . . . . . . . . . . 15 (𝑚 = 𝑦 𝑚 = 𝑦)
4039sseq2d 3976 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑦))
4140rexbidv 3175 . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
4443anbi2d 629 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
45 inteq 4910 . . . . . . . . . . . . . . 15 (𝑚 = (𝑦 ∪ {𝑛}) → 𝑚 = (𝑦 ∪ {𝑛}))
4645sseq2d 3976 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4746rexbidv 3175 . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
5049anbi2d 629 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
51 inteq 4910 . . . . . . . . . . . . . . 15 (𝑚 = 𝑟 𝑚 = 𝑟)
5251sseq2d 3976 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑟))
5352rexbidv 3175 . . . . . . . . . . . . 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 12766 . . . . . . . . . . . . . . . 16 :ℤ⟶𝒫 ℤ
56 ffn 6668 . . . . . . . . . . . . . . . 16 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5755, 56ax-mp 5 . . . . . . . . . . . . . . 15 Fn ℤ
58 0z 12510 . . . . . . . . . . . . . . 15 0 ∈ ℤ
59 fnfvelrn 7031 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 0 ∈ ℤ) → (ℤ‘0) ∈ ran ℤ)
6057, 58, 59mp2an 690 . . . . . . . . . . . . . 14 (ℤ‘0) ∈ ran ℤ
61 ssv 3968 . . . . . . . . . . . . . . 15 (𝐹 “ (ℤ‘0)) ⊆ V
62 int0 4923 . . . . . . . . . . . . . . 15 ∅ = V
6361, 62sseqtrri 3981 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ‘0)) ⊆
64 imaeq2 6009 . . . . . . . . . . . . . . . 16 (𝑘 = (ℤ‘0) → (𝐹𝑘) = (𝐹 “ (ℤ‘0)))
6564sseq1d 3975 . . . . . . . . . . . . . . 15 (𝑘 = (ℤ‘0) → ((𝐹𝑘) ⊆ ∅ ↔ (𝐹 “ (ℤ‘0)) ⊆ ∅))
6665rspcev 3581 . . . . . . . . . . . . . 14 (((ℤ‘0) ∈ ran ℤ ∧ (𝐹 “ (ℤ‘0)) ⊆ ∅) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
6760, 63, 66mp2an 690 . . . . . . . . . . . . 13 𝑘 ∈ ran ℤ(𝐹𝑘) ⊆
6867a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
69 ssun1 4132 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑛})
70 ssralv 4010 . . . . . . . . . . . . . . . . 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 617 . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . . 18 {𝑛} ⊆ (𝑦 ∪ {𝑛})
75 ssralv 4010 . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
78 eqeq1 2740 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
7978rexbidv 3175 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
8077, 79ralsn 4642 . . . . . . . . . . . . . . . . 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 15229 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ) → (𝑢𝑘) ∈ ran ℤ)
838, 10sstrid 3955 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐹𝑢) ⊆ 𝑋)
8483, 12sseqtrd 3984 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
8515sscls 22407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
867, 84, 85syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
87 sseq2 3970 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → ((𝐹𝑢) ⊆ 𝑛 ↔ (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢))))
8886, 87syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝐹𝑢) ⊆ 𝑛))
89 inss2 4189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑘
90 inss1 4188 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑢
91 imass2 6054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑘 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘))
92 imass2 6054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑢 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢))
9391, 92anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → ((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)))
94 ssin 4190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢))
97 ss2in 4196 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → ((𝐹𝑘) ∩ (𝐹𝑢)) ⊆ ( 𝑦𝑛))
9896, 97sstrid 3955 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ ( 𝑦𝑛))
9977intunsn 4950 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∪ {𝑛}) = ( 𝑦𝑛)
10098, 99sseqtrrdi 3995 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛}))
101100expcom 414 . . . . . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
104 imaeq2 6009 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = (𝑢𝑘) → (𝐹𝑚) = (𝐹 “ (𝑢𝑘)))
105104sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑢𝑘) → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
106105rspcev 3581 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢𝑘) ∈ ran ℤ ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}))
107106expcom 414 . . . . . . . . . . . . . . . . . . . . . 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 3204 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ(𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛})))
112 reeanv 3217 . . . . . . . . . . . . . . . . . 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 6009 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
114113sseq1d 3975 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑘 → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
115114cbvrexvw 3226 . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 9108 . . . . . . . . . . 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 455 . . . . . . . . 9 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟)
1259ffnd 6669 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
126 inss1 4188 . . . . . . . . . . . . . . 15 (𝑘 ∩ ℕ) ⊆ 𝑘
127 imass2 6054 . . . . . . . . . . . . . . 15 ((𝑘 ∩ ℕ) ⊆ 𝑘 → (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘))
128126, 127ax-mp 5 . . . . . . . . . . . . . 14 (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘)
129 nnuz 12806 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
130 1z 12533 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
131 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . . . 21 ((ℤ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ‘1) ∈ ran ℤ)
13257, 130, 131mp2an 690 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘1) ∈ ran ℤ
133129, 132eqeltri 2834 . . . . . . . . . . . . . . . . . . 19 ℕ ∈ ran ℤ
134 uzin2 15229 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ran ℤ ∧ ℕ ∈ ran ℤ) → (𝑘 ∩ ℕ) ∈ ran ℤ)
135133, 134mpan2 689 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ∈ ran ℤ)
136 uzn0 12780 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∩ ℕ) ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
137135, 136syl 17 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
138 n0 4306 . . . . . . . . . . . . . . . . 17 ((𝑘 ∩ ℕ) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
139137, 138sylib 217 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran ℤ → ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
140 fnfun 6602 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → Fun 𝐹)
141 inss2 4189 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∩ ℕ) ⊆ ℕ
142 fndm 6605 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn ℕ → dom 𝐹 = ℕ)
143141, 142sseqtrrid 3997 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → (𝑘 ∩ ℕ) ⊆ dom 𝐹)
144 funfvima2 7181 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (𝑘 ∩ ℕ) ⊆ dom 𝐹) → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
145140, 143, 144syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ))))
146 ne0i 4294 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
147145, 146syl6 35 . . . . . . . . . . . . . . . . 17 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
148147exlimdv 1936 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 14 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
151 ssn0 4360 . . . . . . . . . . . . . 14 (((𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅) → (𝐹𝑘) ≠ ∅)
152128, 150, 151sylancr 587 . . . . . . . . . . . . 13 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹𝑘) ≠ ∅)
153 ssn0 4360 . . . . . . . . . . . . . 14 (((𝐹𝑘) ⊆ 𝑟 ∧ (𝐹𝑘) ≠ ∅) → 𝑟 ≠ ∅)
154153expcom 414 . . . . . . . . . . . . 13 ((𝐹𝑘) ≠ ∅ → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
155152, 154syl 17 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → ((𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
156155rexlimdva 3152 . . . . . . . . . . 11 (𝐹 Fn ℕ → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
157125, 156syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ 𝑟 𝑟 ≠ ∅))
158157adantr 481 . . . . . . . . 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 2999 . . . . . . 7 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∅ ≠ 𝑟)
161160neneqd 2948 . . . . . 6 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ¬ ∅ = 𝑟)
16230, 161sylan2b 594 . . . . 5 ((𝜑𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)) → ¬ ∅ = 𝑟)
163162nrexdv 3146 . . . 4 (𝜑 → ¬ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
164 0ex 5264 . . . . 5 ∅ ∈ V
165 zex 12508 . . . . . . . 8 ℤ ∈ V
166165pwex 5335 . . . . . . 7 𝒫 ℤ ∈ V
167 frn 6675 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
16855, 167ax-mp 5 . . . . . . 7 ran ℤ ⊆ 𝒫 ℤ
169166, 168ssexi 5279 . . . . . 6 ran ℤ ∈ V
170169abrexex 7895 . . . . 5 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ V
171 elfi 9349 . . . . 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 690 . . . 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 22746 . . . . . 6 (𝐽 ∈ Comp → 𝐽 ∈ Top)
175 cmpfi 22759 . . . . . 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 6842 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (fi‘𝑚) = (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
179178eleq2d 2823 . . . . . . 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 4910 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → 𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
182181neeq1d 3003 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅))
183 n0 4306 . . . . . . 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 3578 . . . 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 22581 . . 3 Rel (⇝𝑡𝐽)
190 r19.23v 3179 . . . . . 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 6855 . . . . . . . 8 ((cls‘𝐽)‘(𝐹𝑢)) ∈ V
193 eleq2 2826 . . . . . . . 8 (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝑦𝑘𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))))
194192, 193ceqsalv 3481 . . . . . . 7 (∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
195194ralbii 3096 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
196 ralcom4 3269 . . . . . 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 3449 . . . . . 6 𝑦 ∈ V
199198elintab 4919 . . . . 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 2736 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))
202 imaeq2 6009 . . . . . . . . . . . . 13 (𝑢 = ℕ → (𝐹𝑢) = (𝐹 “ ℕ))
203202fveq2d 6846 . . . . . . . . . . . 12 (𝑢 = ℕ → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ ℕ)))
204203rspceeqv 3595 . . . . . . . . . . 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 690 . . . . . . . . . 10 𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))
206 fvex 6855 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ V
207 eqeq1 2740 . . . . . . . . . . . 12 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
208207rexbidv 3175 . . . . . . . . . . 11 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
209206, 208elab 3630 . . . . . . . . . 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 4924 . . . . . . . . 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 6024 . . . . . . . . . . 11 (𝐹 “ ℕ) ⊆ ran 𝐹
214213, 13sstrid 3955 . . . . . . . . . 10 (𝜑 → (𝐹 “ ℕ) ⊆ 𝐽)
21515clsss3 22410 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹 “ ℕ) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
2167, 214, 215syl2anc 584 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
217216, 12sseqtrrd 3985 . . . . . . . 8 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝑋)
218212, 217sstrid 3955 . . . . . . 7 (𝜑 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ 𝑋)
219218sselda 3944 . . . . . 6 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝑦𝑋)
220200, 219sylan2b 594 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝑦𝑋)
221 heibor1.5 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (Cau‘𝐷))
222 1zzd 12534 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
223129, 4, 222iscau3 24642 . . . . . . . . . . . 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 496 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦))
226 simp3 1138 . . . . . . . . . . . . 13 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
227226ralimi 3086 . . . . . . . . . . . 12 (∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
228227reximi 3087 . . . . . . . . . . 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 3086 . . . . . . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑦 ∈ ℝ+𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
232 rphalfcl 12942 . . . . . . . 8 (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
233 breq2 5109 . . . . . . . . . . 11 (𝑦 = (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
2342332ralbidv 3212 . . . . . . . . . 10 (𝑦 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
235234rexbidv 3175 . . . . . . . . 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 3580 . . . . . . . 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 596 . . . . . . 7 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2))
2389ffund 6672 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
239238ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → Fun 𝐹)
2407ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐽 ∈ Top)
241 imassrn 6024 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ𝑚)) ⊆ ran 𝐹
242241, 13sstrid 3955 . . . . . . . . . . . . 13 (𝜑 → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
243242ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
244 nnz 12520 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
245 fnfvelrn 7031 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 𝑚 ∈ ℤ) → (ℤ𝑚) ∈ ran ℤ)
24657, 244, 245sylancr 587 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ∈ ran ℤ)
247246ad2antll 727 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ∈ ran ℤ)
248 simplr 767 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
249 imaeq2 6009 . . . . . . . . . . . . . . . 16 (𝑢 = (ℤ𝑚) → (𝐹𝑢) = (𝐹 “ (ℤ𝑚)))
250249fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑢 = (ℤ𝑚) → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚))))
251250eleq2d 2823 . . . . . . . . . . . . . 14 (𝑢 = (ℤ𝑚) → (𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))))
252251rspcv 3577 . . . . . . . . . . . . 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 724 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐷 ∈ (∞Met‘𝑋))
255220adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦𝑋)
256232ad2antrl 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ+)
257256rpxrd 12958 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ*)
2585blopn 23856 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ*) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
259254, 255, 257, 258syl3anc 1371 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
260 blcntr 23766 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
261254, 255, 256, 260syl3anc 1371 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
26215clsndisj 22426 . . . . . . . . . . . 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 1378 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅)
264 n0 4306 . . . . . . . . . . . 12 (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))))
265 inss2 4189 . . . . . . . . . . . . . . . . 17 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝐹 “ (ℤ𝑚))
266265sseli 3940 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝐹 “ (ℤ𝑚)))
267 fvelima 6908 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑛 ∈ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
268266, 267sylan2 593 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) = 𝑛)
269 inss1 4188 . . . . . . . . . . . . . . . . . . 19 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝑦(ball‘𝐷)(𝑟 / 2))
270269sseli 3940 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
271270adantl 482 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚)))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
272 eleq1a 2833 . . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
277276exlimdv 1936 . . . . . . . . . . . 12 (Fun 𝐹 → (∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
278264, 277biimtrid 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 3117 . . . . . . . . . . 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 12820 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ⊆ ℕ)
282281ad2antll 727 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ⊆ ℕ)
283 simprlr 778 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
2844ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
285 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . 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 12958 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ*)
288 simpllr 774 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑦𝑋)
2899ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐹:ℕ⟶𝑋)
290 eluznn 12843 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑚)) → 𝑘 ∈ ℕ)
291290ad2ant2lr 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑟 ∈ ℝ+𝑚 ∈ ℕ) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → 𝑘 ∈ ℕ)
292291ad2ant2lr 746 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑘 ∈ ℕ)
293289, 292ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ 𝑋)
294 elbl3 23745 . . . . . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2))
2972ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (Met‘𝑋))
298 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ (ℤ𝑘))
299 eluznn 12843 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (ℤ𝑘)) → 𝑛 ∈ ℕ)
300292, 298, 299syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ ℕ)
301289, 300ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑛) ∈ 𝑋)
302 metcl 23685 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
303297, 293, 301, 302syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
304 metcl 23685 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋𝑦𝑋) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
305297, 293, 288, 304syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
306286rpred 12957 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ)
307 lt2add 11640 . . . . . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . . . . 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 12959 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℂ)
3113102halvesd 12399 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝑟 / 2) + (𝑟 / 2)) = 𝑟)
312311breq2d 5117 . . . . . . . . . . . . . . . . . . . . 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 23694 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋)) → ((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)))
315297, 293, 301, 288, 314syl13anc 1372 . . . . . . . . . . . . . . . . . . . . 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 23685 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
317297, 301, 288, 316syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
318303, 305readdcld 11184 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∈ ℝ)
319285rpred 12957 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℝ)
320 lelttr 11245 . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . 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 4011 . . . . . . . . . . . . 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 602 . . . . . . . . . . 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 692 . . . . . . . . 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 468 . . . . . . . 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 3152 . . . . . . 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 3143 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)
339 eqidd 2737 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = (𝐹𝑛))
3405, 4, 129, 222, 339, 9lmmbrf 24626 . . . . . 6 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
341340adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝑦𝑋 ∧ ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)))
342220, 338, 341mpbir2and 711 . . . 4 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → 𝐹(⇝𝑡𝐽)𝑦)
343200, 342sylan2br 595 . . 3 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹(⇝𝑡𝐽)𝑦)
344 releldm 5899 . . 3 ((Rel (⇝𝑡𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ dom (⇝𝑡𝐽))
345189, 343, 344sylancr 587 . 2 ((𝜑𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) → 𝐹 ∈ dom (⇝𝑡𝐽))
346188, 345exlimddv 1938 1 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   cint 4907   class class class wbr 5105  dom cdm 5633  ran crn 5634  cima 5636  Rel wrel 5638  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  pm cpm 8766  Fincfn 8883  ficfi 9346  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189  cle 11190   / cdiv 11812  cn 12153  2c2 12208  cz 12499  cuz 12763  +crp 12915  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  Topctop 22242  Clsdccld 22367  clsccl 22369  𝑡clm 22577  Compccmp 22737  Cauccau 24617
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-lm 22580  df-cmp 22738  df-cau 24620
This theorem is referenced by:  heibor1  36269
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