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Theorem heiborlem3 33742
Description: Lemma for heibor 33750. Using countable choice ax-cc 9295, we have fixed in advance a collection of finite 2↑-𝑛 nets (𝐹𝑛) for 𝑋 (note that an 𝑟-net is a set of points in 𝑋 whose 𝑟 -balls cover 𝑋). The set 𝐺 is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set 𝐾). If the theorem was false, then 𝑋 would be in 𝐾, and so some ball at each level would also be in 𝐾. But we can say more than this; given a ball (𝑦𝐵𝑛) on level 𝑛, since level 𝑛 + 1 covers the space and thus also (𝑦𝐵𝑛), using heiborlem1 33740 there is a ball on the next level whose intersection with (𝑦𝐵𝑛) also has no finite subcover. Now since the set 𝐺 is a countable union of finite sets, it is countable (which needs ax-cc 9295 via iunctb 9434), and so we can apply ax-cc 9295 to 𝐺 directly to get a function from 𝐺 to itself, which points from each ball in 𝐾 to a ball on the next level in 𝐾, and such that the intersection between these balls is also in 𝐾. (Contributed by Jeff Madsen, 18-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heibor.4 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6 (𝜑𝐷 ∈ (CMet‘𝑋))
heibor.7 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
Assertion
Ref Expression
heiborlem3 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
Distinct variable groups:   𝑥,𝑛,𝑦,𝑢,𝐹   𝑥,𝑔,𝐺   𝜑,𝑔,𝑥   𝑔,𝑚,𝑛,𝑢,𝑣,𝑦,𝑧,𝐷,𝑥   𝐵,𝑔,𝑛,𝑢,𝑣,𝑦   𝑔,𝐽,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑔,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑔,𝑋,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑔,𝐾,𝑛,𝑥,𝑦,𝑧   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑔,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)

Proof of Theorem heiborlem3
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 nn0ex 11336 . . . . . 6 0 ∈ V
2 fvex 6239 . . . . . . 7 (𝐹𝑡) ∈ V
3 snex 4938 . . . . . . 7 {𝑡} ∈ V
42, 3xpex 7004 . . . . . 6 ((𝐹𝑡) × {𝑡}) ∈ V
51, 4iunex 7189 . . . . 5 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V
6 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
76relopabi 5278 . . . . . . . 8 Rel 𝐺
8 1st2nd 7258 . . . . . . . 8 ((Rel 𝐺𝑥𝐺) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
97, 8mpan 706 . . . . . . 7 (𝑥𝐺𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109eleq1d 2715 . . . . . . . . . . 11 (𝑥𝐺 → (𝑥𝐺 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺))
11 df-br 4686 . . . . . . . . . . 11 ((1st𝑥)𝐺(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺)
1210, 11syl6bbr 278 . . . . . . . . . 10 (𝑥𝐺 → (𝑥𝐺 ↔ (1st𝑥)𝐺(2nd𝑥)))
13 heibor.1 . . . . . . . . . . 11 𝐽 = (MetOpen‘𝐷)
14 heibor.3 . . . . . . . . . . 11 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
15 fvex 6239 . . . . . . . . . . 11 (1st𝑥) ∈ V
16 fvex 6239 . . . . . . . . . . 11 (2nd𝑥) ∈ V
1713, 14, 6, 15, 16heiborlem2 33741 . . . . . . . . . 10 ((1st𝑥)𝐺(2nd𝑥) ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
1812, 17syl6bb 276 . . . . . . . . 9 (𝑥𝐺 → (𝑥𝐺 ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)))
1918ibi 256 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
2016snid 4241 . . . . . . . . . . . 12 (2nd𝑥) ∈ {(2nd𝑥)}
21 opelxp 5180 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ ((1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ (2nd𝑥) ∈ {(2nd𝑥)}))
2220, 21mpbiran2 974 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
23 fveq2 6229 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → (𝐹𝑡) = (𝐹‘(2nd𝑥)))
24 sneq 4220 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → {𝑡} = {(2nd𝑥)})
2523, 24xpeq12d 5174 . . . . . . . . . . . . 13 (𝑡 = (2nd𝑥) → ((𝐹𝑡) × {𝑡}) = ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}))
2625eleq2d 2716 . . . . . . . . . . . 12 (𝑡 = (2nd𝑥) → (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})))
2726rspcev 3340 . . . . . . . . . . 11 (((2nd𝑥) ∈ ℕ0 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
2822, 27sylan2br 492 . . . . . . . . . 10 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
29 eliun 4556 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ↔ ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
3028, 29sylibr 224 . . . . . . . . 9 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
31303adant3 1101 . . . . . . . 8 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3219, 31syl 17 . . . . . . 7 (𝑥𝐺 → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
339, 32eqeltrd 2730 . . . . . 6 (𝑥𝐺𝑥 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3433ssriv 3640 . . . . 5 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
35 ssdomg 8043 . . . . 5 ( 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V → (𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) → 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})))
365, 34, 35mp2 9 . . . 4 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
37 nn0ennn 12818 . . . . . . 7 0 ≈ ℕ
38 nnenom 12819 . . . . . . 7 ℕ ≈ ω
3937, 38entri 8051 . . . . . 6 0 ≈ ω
40 endom 8024 . . . . . 6 (ℕ0 ≈ ω → ℕ0 ≼ ω)
4139, 40ax-mp 5 . . . . 5 0 ≼ ω
42 vex 3234 . . . . . . . 8 𝑡 ∈ V
432, 42xpsnen 8085 . . . . . . 7 ((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡)
44 inss2 3867 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
45 heibor.7 . . . . . . . . . 10 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
4645ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ (𝒫 𝑋 ∩ Fin))
4744, 46sseldi 3634 . . . . . . . 8 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ Fin)
48 isfinite 8587 . . . . . . . . 9 ((𝐹𝑡) ∈ Fin ↔ (𝐹𝑡) ≺ ω)
49 sdomdom 8025 . . . . . . . . 9 ((𝐹𝑡) ≺ ω → (𝐹𝑡) ≼ ω)
5048, 49sylbi 207 . . . . . . . 8 ((𝐹𝑡) ∈ Fin → (𝐹𝑡) ≼ ω)
5147, 50syl 17 . . . . . . 7 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ≼ ω)
52 endomtr 8055 . . . . . . 7 ((((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡) ∧ (𝐹𝑡) ≼ ω) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5343, 51, 52sylancr 696 . . . . . 6 ((𝜑𝑡 ∈ ℕ0) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5453ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
55 iunctb 9434 . . . . 5 ((ℕ0 ≼ ω ∧ ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
5641, 54, 55sylancr 696 . . . 4 (𝜑 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
57 domtr 8050 . . . 4 ((𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∧ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝐺 ≼ ω)
5836, 56, 57sylancr 696 . . 3 (𝜑𝐺 ≼ ω)
5919simp1d 1093 . . . . . . . . 9 (𝑥𝐺 → (2nd𝑥) ∈ ℕ0)
60 peano2nn0 11371 . . . . . . . . 9 ((2nd𝑥) ∈ ℕ0 → ((2nd𝑥) + 1) ∈ ℕ0)
6159, 60syl 17 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) + 1) ∈ ℕ0)
62 ffvelrn 6397 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6345, 61, 62syl2an 493 . . . . . . 7 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6444, 63sseldi 3634 . . . . . 6 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ Fin)
65 iunin2 4616 . . . . . . . 8 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
66 heibor.8 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
67 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
6867cbviunv 4591 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛)
69 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = ((2nd𝑥) + 1) → (𝐹𝑛) = (𝐹‘((2nd𝑥) + 1)))
7069iuneq1d 4577 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
7168, 70syl5eq 2697 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
72 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd𝑥) + 1)))
7372iuneq2d 4579 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7471, 73eqtrd 2685 . . . . . . . . . . . . 13 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7574eqeq2d 2661 . . . . . . . . . . . 12 (𝑛 = ((2nd𝑥) + 1) → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
7675rspccva 3339 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7766, 61, 76syl2an 493 . . . . . . . . . 10 ((𝜑𝑥𝐺) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7877ineq2d 3847 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
799fveq2d 6233 . . . . . . . . . . . . . 14 (𝑥𝐺 → (𝐵𝑥) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩))
80 df-ov 6693 . . . . . . . . . . . . . 14 ((1st𝑥)𝐵(2nd𝑥)) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩)
8179, 80syl6eqr 2703 . . . . . . . . . . . . 13 (𝑥𝐺 → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
8281adantl 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
83 inss1 3866 . . . . . . . . . . . . . . . 16 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84 ffvelrn 6397 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (2nd𝑥) ∈ ℕ0) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8545, 59, 84syl2an 493 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8683, 85sseldi 3634 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ 𝒫 𝑋)
8786elpwid 4203 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ⊆ 𝑋)
8819simp2d 1094 . . . . . . . . . . . . . . 15 (𝑥𝐺 → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
8988adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
9087, 89sseldd 3637 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1st𝑥) ∈ 𝑋)
9159adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (2nd𝑥) ∈ ℕ0)
92 oveq1 6697 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑚 = (2nd𝑥) → (2↑𝑚) = (2↑(2nd𝑥)))
9493oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑚 = (2nd𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd𝑥))))
9594oveq2d 6706 . . . . . . . . . . . . . 14 (𝑚 = (2nd𝑥) → ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
96 heibor.5 . . . . . . . . . . . . . 14 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
97 ovex 6718 . . . . . . . . . . . . . 14 ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ∈ V
9892, 95, 96, 97ovmpt2 6838 . . . . . . . . . . . . 13 (((1st𝑥) ∈ 𝑋 ∧ (2nd𝑥) ∈ ℕ0) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
9990, 91, 98syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
10082, 99eqtrd 2685 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
101 heibor.6 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (CMet‘𝑋))
102 cmetmet 23130 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103101, 102syl 17 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ (Met‘𝑋))
104 metxmet 22186 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105103, 104syl 17 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (∞Met‘𝑋))
106105adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107 2nn 11223 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
108 nnexpcl 12913 . . . . . . . . . . . . . . . 16 ((2 ∈ ℕ ∧ (2nd𝑥) ∈ ℕ0) → (2↑(2nd𝑥)) ∈ ℕ)
109107, 91, 108sylancr 696 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℕ)
110109nnrpd 11908 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℝ+)
111110rpreccld 11920 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ+)
112111rpxrd 11911 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ*)
113 blssm 22270 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑥) ∈ 𝑋 ∧ (1 / (2↑(2nd𝑥))) ∈ ℝ*) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
114106, 90, 112, 113syl3anc 1366 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
115100, 114eqsstrd 3672 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑋)
116 df-ss 3621 . . . . . . . . . 10 ((𝐵𝑥) ⊆ 𝑋 ↔ ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
117115, 116sylib 208 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
11878, 117eqtr3d 2687 . . . . . . . 8 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
11965, 118syl5eq 2697 . . . . . . 7 ((𝜑𝑥𝐺) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
120 eqimss2 3691 . . . . . . 7 ( 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥) → (𝐵𝑥) ⊆ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))))
121119, 120syl 17 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))))
12219simp3d 1095 . . . . . . . 8 (𝑥𝐺 → ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)
12381, 122eqeltrd 2730 . . . . . . 7 (𝑥𝐺 → (𝐵𝑥) ∈ 𝐾)
124123adantl 481 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ∈ 𝐾)
125 fvex 6239 . . . . . . . 8 (𝐵𝑥) ∈ V
126125inex1 4832 . . . . . . 7 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ V
12713, 14, 126heiborlem1 33740 . . . . . 6 (((𝐹‘((2nd𝑥) + 1)) ∈ Fin ∧ (𝐵𝑥) ⊆ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∧ (𝐵𝑥) ∈ 𝐾) → ∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
12864, 121, 124, 127syl3anc 1366 . . . . 5 ((𝜑𝑥𝐺) → ∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
12983, 63sseldi 3634 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ 𝒫 𝑋)
130129elpwid 4203 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝑋)
13113mopnuni 22293 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
132105, 131syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = 𝐽)
133132adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → 𝑋 = 𝐽)
134130, 133sseqtrd 3674 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝐽)
135134sselda 3636 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))) → 𝑡 𝐽)
136135adantrr 753 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡 𝐽)
13761adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐺) → ((2nd𝑥) + 1) ∈ ℕ0)
138 id 22 . . . . . . . . . 10 (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)))
139 snfi 8079 . . . . . . . . . . . 12 {(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin
140 inss2 3867 . . . . . . . . . . . . 13 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ (𝑡𝐵((2nd𝑥) + 1))
141 ovex 6718 . . . . . . . . . . . . . . 15 (𝑡𝐵((2nd𝑥) + 1)) ∈ V
142141unisn 4483 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = (𝑡𝐵((2nd𝑥) + 1))
143 uniiun 4605 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
144142, 143eqtr3i 2675 . . . . . . . . . . . . 13 (𝑡𝐵((2nd𝑥) + 1)) = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
145140, 144sseqtri 3670 . . . . . . . . . . . 12 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
146 vex 3234 . . . . . . . . . . . . 13 𝑔 ∈ V
14713, 14, 146heiborlem1 33740 . . . . . . . . . . . 12 (({(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔 ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾)
148139, 145, 147mp3an12 1454 . . . . . . . . . . 11 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → ∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾)
149 eleq1 2718 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐵((2nd𝑥) + 1)) → (𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
150141, 149rexsn 4255 . . . . . . . . . . 11 (∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
151148, 150sylib 208 . . . . . . . . . 10 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
152 ovex 6718 . . . . . . . . . . . 12 ((2nd𝑥) + 1) ∈ V
15313, 14, 6, 42, 152heiborlem2 33741 . . . . . . . . . . 11 (𝑡𝐺((2nd𝑥) + 1) ↔ (((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
154153biimpri 218 . . . . . . . . . 10 ((((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
155137, 138, 151, 154syl3an 1408 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
1561553expb 1285 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡𝐺((2nd𝑥) + 1))
157 simprr 811 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
158136, 156, 157jca32 557 . . . . . . 7 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
159158ex 449 . . . . . 6 ((𝜑𝑥𝐺) → ((𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))))
160159reximdv2 3043 . . . . 5 ((𝜑𝑥𝐺) → (∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → ∃𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
161128, 160mpd 15 . . . 4 ((𝜑𝑥𝐺) → ∃𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
162161ralrimiva 2995 . . 3 (𝜑 → ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
163 fvex 6239 . . . . . 6 (MetOpen‘𝐷) ∈ V
16413, 163eqeltri 2726 . . . . 5 𝐽 ∈ V
165164uniex 6995 . . . 4 𝐽 ∈ V
166 breq1 4688 . . . . 5 (𝑡 = (𝑔𝑥) → (𝑡𝐺((2nd𝑥) + 1) ↔ (𝑔𝑥)𝐺((2nd𝑥) + 1)))
167 oveq1 6697 . . . . . . 7 (𝑡 = (𝑔𝑥) → (𝑡𝐵((2nd𝑥) + 1)) = ((𝑔𝑥)𝐵((2nd𝑥) + 1)))
168167ineq2d 3847 . . . . . 6 (𝑡 = (𝑔𝑥) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))))
169168eleq1d 2715 . . . . 5 (𝑡 = (𝑔𝑥) → (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
170166, 169anbi12d 747 . . . 4 (𝑡 = (𝑔𝑥) → ((𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
171165, 170axcc4dom 9301 . . 3 ((𝐺 ≼ ω ∧ ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
17258, 162, 171syl2anc 694 . 2 (𝜑 → ∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
173 exsimpr 1836 . 2 (∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
174172, 173syl 17 1 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  {copab 4745   × cxp 5141  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  1st c1st 7208  2nd c2nd 7209  cen 7994  cdom 7995  csdm 7996  Fincfn 7997  1c1 9975   + caddc 9977  *cxr 10111   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cexp 12900  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  CMetcms 23098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-seq 12842  df-exp 12901  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmet 23101
This theorem is referenced by:  heiborlem10  33749
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