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Theorem heibor1lem 37816
Description: Lemma for heibor1 37817. 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 24344 . . . . . . . . . 10 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
42, 3syl 17 . . . . . . . . 9 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 heibor.1 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
65mopntop 24450 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Top)
8 imassrn 6089 . . . . . . . . 9 (𝐹𝑢) ⊆ ran 𝐹
9 heibor1.6 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶𝑋)
109frnd 6744 . . . . . . . . . 10 (𝜑 → ran 𝐹𝑋)
115mopnuni 24451 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
124, 11syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1310, 12sseqtrd 4020 . . . . . . . . 9 (𝜑 → ran 𝐹 𝐽)
148, 13sstrid 3995 . . . . . . . 8 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
15 eqid 2737 . . . . . . . . 9 𝐽 = 𝐽
1615clscld 23055 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
177, 14, 16syl2anc 584 . . . . . . 7 (𝜑 → ((cls‘𝐽)‘(𝐹𝑢)) ∈ (Clsd‘𝐽))
18 eleq1a 2836 . . . . . . 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 3160 . . . . 5 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑘 ∈ (Clsd‘𝐽)))
2120abssdv 4068 . . . 4 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
22 fvex 6919 . . . . 5 (Clsd‘𝐽) ∈ V
2322elpw2 5334 . . . 4 ({𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽) ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ (Clsd‘𝐽))
2421, 23sylibr 234 . . 3 (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ 𝒫 (Clsd‘𝐽))
25 elin 3967 . . . . . . 7 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∧ 𝑟 ∈ Fin))
26 velpw 4605 . . . . . . . . 9 (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ 𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
27 ssabral 4065 . . . . . . . . 9 (𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))
2826, 27bitri 275 . . . . . . . 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 275 . . . . . 6 (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin) ↔ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin))
31 raleq 3323 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3231anbi2d 630 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
33 inteq 4949 . . . . . . . . . . . . . . 15 (𝑚 = ∅ → 𝑚 = ∅)
3433sseq2d 4016 . . . . . . . . . . . . . 14 (𝑚 = ∅ → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ ∅))
3534rexbidv 3179 . . . . . . . . . . . . 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 3323 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
3837anbi2d 630 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑦𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
39 inteq 4949 . . . . . . . . . . . . . . 15 (𝑚 = 𝑦 𝑚 = 𝑦)
4039sseq2d 4016 . . . . . . . . . . . . . 14 (𝑚 = 𝑦 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑦))
4140rexbidv 3179 . . . . . . . . . . . . 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 3323 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
4443anbi2d 630 . . . . . . . . . . . . 13 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
45 inteq 4949 . . . . . . . . . . . . . . 15 (𝑚 = (𝑦 ∪ {𝑛}) → 𝑚 = (𝑦 ∪ {𝑛}))
4645sseq2d 4016 . . . . . . . . . . . . . 14 (𝑚 = (𝑦 ∪ {𝑛}) → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
4746rexbidv 3179 . . . . . . . . . . . . 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 3323 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → (∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))))
5049anbi2d 630 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → ((𝜑 ∧ ∀𝑘𝑚𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) ↔ (𝜑 ∧ ∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)))))
51 inteq 4949 . . . . . . . . . . . . . . 15 (𝑚 = 𝑟 𝑚 = 𝑟)
5251sseq2d 4016 . . . . . . . . . . . . . 14 (𝑚 = 𝑟 → ((𝐹𝑘) ⊆ 𝑚 ↔ (𝐹𝑘) ⊆ 𝑟))
5352rexbidv 3179 . . . . . . . . . . . . 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 12881 . . . . . . . . . . . . . . . 16 :ℤ⟶𝒫 ℤ
56 ffn 6736 . . . . . . . . . . . . . . . 16 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5755, 56ax-mp 5 . . . . . . . . . . . . . . 15 Fn ℤ
58 0z 12624 . . . . . . . . . . . . . . 15 0 ∈ ℤ
59 fnfvelrn 7100 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 0 ∈ ℤ) → (ℤ‘0) ∈ ran ℤ)
6057, 58, 59mp2an 692 . . . . . . . . . . . . . 14 (ℤ‘0) ∈ ran ℤ
61 ssv 4008 . . . . . . . . . . . . . . 15 (𝐹 “ (ℤ‘0)) ⊆ V
62 int0 4962 . . . . . . . . . . . . . . 15 ∅ = V
6361, 62sseqtrri 4033 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ‘0)) ⊆
64 imaeq2 6074 . . . . . . . . . . . . . . . 16 (𝑘 = (ℤ‘0) → (𝐹𝑘) = (𝐹 “ (ℤ‘0)))
6564sseq1d 4015 . . . . . . . . . . . . . . 15 (𝑘 = (ℤ‘0) → ((𝐹𝑘) ⊆ ∅ ↔ (𝐹 “ (ℤ‘0)) ⊆ ∅))
6665rspcev 3622 . . . . . . . . . . . . . 14 (((ℤ‘0) ∈ ran ℤ ∧ (𝐹 “ (ℤ‘0)) ⊆ ∅) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
6760, 63, 66mp2an 692 . . . . . . . . . . . . 13 𝑘 ∈ ran ℤ(𝐹𝑘) ⊆
6867a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))) → ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ ∅)
69 ssun1 4178 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑛})
70 ssralv 4052 . . . . . . . . . . . . . . . . 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 4179 . . . . . . . . . . . . . . . . . 18 {𝑛} ⊆ (𝑦 ∪ {𝑛})
75 ssralv 4052 . . . . . . . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
78 eqeq1 2741 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
7978rexbidv 3179 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢))))
8077, 79ralsn 4681 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
8176, 80sylib 218 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → ∃𝑢 ∈ ran ℤ𝑛 = ((cls‘𝐽)‘(𝐹𝑢)))
82 uzin2 15383 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ) → (𝑢𝑘) ∈ ran ℤ)
838, 10sstrid 3995 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐹𝑢) ⊆ 𝑋)
8483, 12sseqtrd 4020 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐹𝑢) ⊆ 𝐽)
8515sscls 23064 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐽 ∈ Top ∧ (𝐹𝑢) ⊆ 𝐽) → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
867, 84, 85syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢)))
87 sseq2 4010 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → ((𝐹𝑢) ⊆ 𝑛 ↔ (𝐹𝑢) ⊆ ((cls‘𝐽)‘(𝐹𝑢))))
8886, 87syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝐹𝑢) ⊆ 𝑛))
89 inss2 4238 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑘
90 inss1 4237 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢𝑘) ⊆ 𝑢
91 imass2 6120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑘 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘))
92 imass2 6120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑘) ⊆ 𝑢 → (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢))
9391, 92anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → ((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)))
94 ssin 4239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑢𝑘)) ⊆ (𝐹𝑢)) ↔ (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9593, 94sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑘) ⊆ 𝑘 ∧ (𝑢𝑘) ⊆ 𝑢) → (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢)))
9689, 90, 95mp2an 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹 “ (𝑢𝑘)) ⊆ ((𝐹𝑘) ∩ (𝐹𝑢))
97 ss2in 4245 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → ((𝐹𝑘) ∩ (𝐹𝑢)) ⊆ ( 𝑦𝑛))
9896, 97sstrid 3995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹𝑘) ⊆ 𝑦 ∧ (𝐹𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢𝑘)) ⊆ ( 𝑦𝑛))
9977intunsn 4987 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∪ {𝑛}) = ( 𝑦𝑛)
10098, 99sseqtrrdi 4025 . . . . . . . . . . . . . . . . . . . . . . . . 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 6074 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = (𝑢𝑘) → (𝐹𝑚) = (𝐹 “ (𝑢𝑘)))
105104sseq1d 4015 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑢𝑘) → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ (𝑢𝑘)) ⊆ (𝑦 ∪ {𝑛})))
106105rspcev 3622 . . . . . . . . . . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑢 ∈ ran ℤ𝑘 ∈ ran ℤ(𝑛 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ (𝐹𝑘) ⊆ 𝑦) → ∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛})))
112 reeanv 3229 . . . . . . . . . . . . . . . . . 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 6074 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
114113sseq1d 4015 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑘 → ((𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ (𝐹𝑘) ⊆ (𝑦 ∪ {𝑛})))
115114cbvrexvw 3238 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ran ℤ(𝐹𝑚) ⊆ (𝑦 ∪ {𝑛}) ↔ ∃𝑘 ∈ ran ℤ(𝐹𝑘) ⊆ (𝑦 ∪ {𝑛}))
116111, 112, 1153imtr3g 295 . . . . . . . . . . . . . . . . 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 9204 . . . . . . . . . . 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 6737 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
126 inss1 4237 . . . . . . . . . . . . . . 15 (𝑘 ∩ ℕ) ⊆ 𝑘
127 imass2 6120 . . . . . . . . . . . . . . 15 ((𝑘 ∩ ℕ) ⊆ 𝑘 → (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘))
128126, 127ax-mp 5 . . . . . . . . . . . . . 14 (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘)
129 nnuz 12921 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℤ‘1)
130 1z 12647 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
131 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . . . 21 ((ℤ Fn ℤ ∧ 1 ∈ ℤ) → (ℤ‘1) ∈ ran ℤ)
13257, 130, 131mp2an 692 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘1) ∈ ran ℤ
133129, 132eqeltri 2837 . . . . . . . . . . . . . . . . . . 19 ℕ ∈ ran ℤ
134 uzin2 15383 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ran ℤ ∧ ℕ ∈ ran ℤ) → (𝑘 ∩ ℕ) ∈ ran ℤ)
135133, 134mpan2 691 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ∈ ran ℤ)
136 uzn0 12895 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∩ ℕ) ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
137135, 136syl 17 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ran ℤ → (𝑘 ∩ ℕ) ≠ ∅)
138 n0 4353 . . . . . . . . . . . . . . . . 17 ((𝑘 ∩ ℕ) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
139137, 138sylib 218 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran ℤ → ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ))
140 fnfun 6668 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → Fun 𝐹)
141 inss2 4238 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∩ ℕ) ⊆ ℕ
142 fndm 6671 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn ℕ → dom 𝐹 = ℕ)
143141, 142sseqtrrid 4027 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℕ → (𝑘 ∩ ℕ) ⊆ dom 𝐹)
144 funfvima2 7251 . . . . . . . . . . . . . . . . . . 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 4341 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅)
147145, 146syl6 35 . . . . . . . . . . . . . . . . 17 (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅))
148147exlimdv 1933 . . . . . . . . . . . . . . . 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 4404 . . . . . . . . . . . . . 14 (((𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹𝑘) ∧ (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅) → (𝐹𝑘) ≠ ∅)
152128, 150, 151sylancr 587 . . . . . . . . . . . . 13 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran ℤ) → (𝐹𝑘) ≠ ∅)
153 ssn0 4404 . . . . . . . . . . . . . 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 3155 . . . . . . . . . . 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 2996 . . . . . . 7 ((𝜑 ∧ (∀𝑘𝑟𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ∧ 𝑟 ∈ Fin)) → ∅ ≠ 𝑟)
161160neneqd 2945 . . . . . 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 3149 . . . 4 (𝜑 → ¬ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
164 0ex 5307 . . . . 5 ∅ ∈ V
165 zex 12622 . . . . . . . 8 ℤ ∈ V
166165pwex 5380 . . . . . . 7 𝒫 ℤ ∈ V
167 frn 6743 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
16855, 167ax-mp 5 . . . . . . 7 ran ℤ ⊆ 𝒫 ℤ
169166, 168ssexi 5322 . . . . . 6 ran ℤ ∈ V
170169abrexex 7987 . . . . 5 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∈ V
171 elfi 9453 . . . . 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 692 . . . 4 (∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ∩ Fin)∅ = 𝑟)
173163, 172sylnibr 329 . . 3 (𝜑 → ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
174 cmptop 23403 . . . . . 6 (𝐽 ∈ Comp → 𝐽 ∈ Top)
175 cmpfi 23416 . . . . . 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 267 . . . 4 (𝐽 ∈ Comp → ∀𝑚 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → 𝑚 ≠ ∅))
178 fveq2 6906 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (fi‘𝑚) = (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}))
179178eleq2d 2827 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (∅ ∈ (fi‘𝑚) ↔ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
180179notbid 318 . . . . . 6 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → (¬ ∅ ∈ (fi‘𝑚) ↔ ¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})))
181 inteq 4949 . . . . . . . 8 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → 𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
182181neeq1d 3000 . . . . . . 7 (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} → ( 𝑚 ≠ ∅ ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅))
183 n0 4353 . . . . . . 7 ( {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ≠ ∅ ↔ ∃𝑦 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
184182, 183bitrdi 287 . . . . . 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 3619 . . . 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 23238 . . 3 Rel (⇝𝑡𝐽)
190 r19.23v 3183 . . . . . 6 (∀𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
191190albii 1819 . . . . 5 (∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
192 fvex 6919 . . . . . . . 8 ((cls‘𝐽)‘(𝐹𝑢)) ∈ V
193 eleq2 2830 . . . . . . . 8 (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → (𝑦𝑘𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))))
194192, 193ceqsalv 3521 . . . . . . 7 (∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
195194ralbii 3093 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
196 ralcom4 3286 . . . . . 6 (∀𝑢 ∈ ran ℤ𝑘(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
197195, 196bitr3i 277 . . . . 5 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∀𝑘𝑢 ∈ ran ℤ(𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
198 vex 3484 . . . . . 6 𝑦 ∈ V
199198elintab 4958 . . . . 5 (𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∀𝑘(∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) → 𝑦𝑘))
200191, 197, 1993bitr4i 303 . . . 4 (∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))})
201 eqid 2737 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))
202 imaeq2 6074 . . . . . . . . . . . . 13 (𝑢 = ℕ → (𝐹𝑢) = (𝐹 “ ℕ))
203202fveq2d 6910 . . . . . . . . . . . 12 (𝑢 = ℕ → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ ℕ)))
204203rspceeqv 3645 . . . . . . . . . . 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 692 . . . . . . . . . 10 𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))
206 fvex 6919 . . . . . . . . . . 11 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ V
207 eqeq1 2741 . . . . . . . . . . . 12 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
208207rexbidv 3179 . . . . . . . . . . 11 (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢)) ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢))))
209206, 208elab 3679 . . . . . . . . . 10 (((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ↔ ∃𝑢 ∈ ran ℤ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹𝑢)))
210205, 209mpbir 231 . . . . . . . . 9 ((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))}
211 intss1 4963 . . . . . . . . 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 6089 . . . . . . . . . . 11 (𝐹 “ ℕ) ⊆ ran 𝐹
214213, 13sstrid 3995 . . . . . . . . . 10 (𝜑 → (𝐹 “ ℕ) ⊆ 𝐽)
21515clsss3 23067 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹 “ ℕ) ⊆ 𝐽) → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
2167, 214, 215syl2anc 584 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝐽)
217216, 12sseqtrrd 4021 . . . . . . . 8 (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝑋)
218212, 217sstrid 3995 . . . . . . 7 (𝜑 {𝑘 ∣ ∃𝑢 ∈ ran ℤ𝑘 = ((cls‘𝐽)‘(𝐹𝑢))} ⊆ 𝑋)
219218sselda 3983 . . . . . 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 12648 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
223129, 4, 222iscau3 25312 . . . . . . . . . . . 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 232 . . . . . . . . . . 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 1139 . . . . . . . . . . . . 13 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
227226ralimi 3083 . . . . . . . . . . . 12 (∀𝑘 ∈ (ℤ𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦) → ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦)
228227reximi 3084 . . . . . . . . . . 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 3083 . . . . . . . . . 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 13062 . . . . . . . 8 (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
233 breq2 5147 . . . . . . . . . . 11 (𝑦 = (𝑟 / 2) → (((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
2342332ralbidv 3221 . . . . . . . . . 10 (𝑦 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ𝑚)∀𝑛 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑛)) < (𝑟 / 2)))
235234rexbidv 3179 . . . . . . . . 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 3621 . . . . . . . 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 6740 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
239238ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → Fun 𝐹)
2407ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝐽 ∈ Top)
241 imassrn 6089 . . . . . . . . . . . . . 14 (𝐹 “ (ℤ𝑚)) ⊆ ran 𝐹
242241, 13sstrid 3995 . . . . . . . . . . . . 13 (𝜑 → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
243242ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝐹 “ (ℤ𝑚)) ⊆ 𝐽)
244 nnz 12634 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
245 fnfvelrn 7100 . . . . . . . . . . . . . . 15 ((ℤ Fn ℤ ∧ 𝑚 ∈ ℤ) → (ℤ𝑚) ∈ ran ℤ)
24657, 244, 245sylancr 587 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ∈ ran ℤ)
247246ad2antll 729 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ∈ ran ℤ)
248 simplr 769 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)))
249 imaeq2 6074 . . . . . . . . . . . . . . . 16 (𝑢 = (ℤ𝑚) → (𝐹𝑢) = (𝐹 “ (ℤ𝑚)))
250249fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑢 = (ℤ𝑚) → ((cls‘𝐽)‘(𝐹𝑢)) = ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚))))
251250eleq2d 2827 . . . . . . . . . . . . . 14 (𝑢 = (ℤ𝑚) → (𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢)) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ𝑚)))))
252251rspcv 3618 . . . . . . . . . . . . 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 726 . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ+)
257256rpxrd 13078 . . . . . . . . . . . . 13 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑟 / 2) ∈ ℝ*)
2585blopn 24513 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ*) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
259254, 255, 257, 258syl3anc 1373 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
260 blcntr 24423 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (𝑟 / 2) ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
261254, 255, 256, 260syl3anc 1373 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
26215clsndisj 23083 . . . . . . . . . . . 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 1380 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅)
264 n0 4353 . . . . . . . . . . . 12 (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))))
265 inss2 4238 . . . . . . . . . . . . . . . . 17 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝐹 “ (ℤ𝑚))
266265sseli 3979 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → 𝑛 ∈ (𝐹 “ (ℤ𝑚)))
267 fvelima 6974 . . . . . . . . . . . . . . . 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 4237 . . . . . . . . . . . . . . . . . . 19 ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) ⊆ (𝑦(ball‘𝐷)(𝑟 / 2))
270269sseli 3979 . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . 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 3170 . . . . . . . . . . . . . . 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 1933 . . . . . . . . . . . 12 (Fun 𝐹 → (∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ𝑚))) → ∃𝑘 ∈ (ℤ𝑚)(𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))))
278264, 277biimtrid 242 . . . . . . . . . . 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 3114 . . . . . . . . . . 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 12937 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (ℤ𝑚) ⊆ ℕ)
282281ad2antll 729 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) → (ℤ𝑚) ⊆ ℕ)
283 simprlr 780 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))
2844ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
285 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . . . . 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 13078 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ*)
288 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑦𝑋)
2899ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐹:ℕ⟶𝑋)
290 eluznn 12960 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑚)) → 𝑘 ∈ ℕ)
291290ad2ant2lr 748 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑟 ∈ ℝ+𝑚 ∈ ℕ) ∧ (𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → 𝑘 ∈ ℕ)
292291ad2ant2lr 748 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑘 ∈ ℕ)
293289, 292ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑘) ∈ 𝑋)
294 elbl3 24402 . . . . . . . . . . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) < (𝑟 / 2))
2972ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝐷 ∈ (Met‘𝑋))
298 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑛 ∈ (ℤ𝑘))
299 eluznn 12960 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝐹𝑛) ∈ 𝑋)
302 metcl 24342 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
303297, 293, 301, 302syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷(𝐹𝑛)) ∈ ℝ)
304 metcl 24342 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑘) ∈ 𝑋𝑦𝑋) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
305297, 293, 288, 304syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑘)𝐷𝑦) ∈ ℝ)
306286rpred 13077 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (𝑟 / 2) ∈ ℝ)
307 lt2add 11748 . . . . . . . . . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . 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 13079 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℂ)
3113102halvesd 12512 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝑟 / 2) + (𝑟 / 2)) = 𝑟)
312311breq2d 5155 . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . 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 24351 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹𝑘) ∈ 𝑋 ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋)) → ((𝐹𝑛)𝐷𝑦) ≤ (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)))
315297, 293, 301, 288, 314syl13anc 1374 . . . . . . . . . . . . . . . . . . . . 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 24342 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹𝑛) ∈ 𝑋𝑦𝑋) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
317297, 301, 288, 316syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → ((𝐹𝑛)𝐷𝑦) ∈ ℝ)
318303, 305readdcld 11290 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → (((𝐹𝑘)𝐷(𝐹𝑛)) + ((𝐹𝑘)𝐷𝑦)) ∈ ℝ)
319285rpred 13077 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑦𝑋) ∧ (𝑟 ∈ ℝ+𝑚 ∈ ℕ)) ∧ ((𝑘 ∈ (ℤ𝑚) ∧ (𝐹𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ𝑘))) → 𝑟 ∈ ℝ)
320 lelttr 11351 . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . 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 4053 . . . . . . . . . . . . 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 694 . . . . . . . . 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 3155 . . . . . . 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 3146 . . . . 5 ((𝜑 ∧ ∀𝑢 ∈ ran ℤ𝑦 ∈ ((cls‘𝐽)‘(𝐹𝑢))) → ∀𝑟 ∈ ℝ+𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ𝑘)((𝐹𝑛)𝐷𝑦) < 𝑟)
339 eqidd 2738 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = (𝐹𝑛))
3405, 4, 129, 222, 339, 9lmmbrf 25296 . . . . . 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 713 . . . 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 5955 . . 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 1935 1 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   cint 4946   class class class wbr 5143  dom cdm 5685  ran crn 5686  cima 5688  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  pm cpm 8867  Fincfn 8985  ficfi 9450  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296   / cdiv 11920  cn 12266  2c2 12321  cz 12613  cuz 12878  +crp 13034  ∞Metcxmet 21349  Metcmet 21350  ballcbl 21351  MetOpencmopn 21354  Topctop 22899  Clsdccld 23024  clsccl 23026  𝑡clm 23234  Compccmp 23394  Cauccau 25287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-lm 23237  df-cmp 23395  df-cau 25290
This theorem is referenced by:  heibor1  37817
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