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Theorem heiborlem3 37821
Description: Lemma for heibor 37829. Using countable choice ax-cc 10476, 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 37819 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 10476 via iunctb 10615), and so we can apply ax-cc 10476 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 12534 . . . . . 6 0 ∈ V
2 fvex 6918 . . . . . . 7 (𝐹𝑡) ∈ V
3 vsnex 5433 . . . . . . 7 {𝑡} ∈ V
42, 3xpex 7774 . . . . . 6 ((𝐹𝑡) × {𝑡}) ∈ V
51, 4iunex 7994 . . . . 5 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V
6 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
76relopabiv 5829 . . . . . . . 8 Rel 𝐺
8 1st2nd 8065 . . . . . . . 8 ((Rel 𝐺𝑥𝐺) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
97, 8mpan 690 . . . . . . 7 (𝑥𝐺𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109eleq1d 2825 . . . . . . . . . . 11 (𝑥𝐺 → (𝑥𝐺 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺))
11 df-br 5143 . . . . . . . . . . 11 ((1st𝑥)𝐺(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺)
1210, 11bitr4di 289 . . . . . . . . . 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 6918 . . . . . . . . . . 11 (1st𝑥) ∈ V
16 fvex 6918 . . . . . . . . . . 11 (2nd𝑥) ∈ V
1713, 14, 6, 15, 16heiborlem2 37820 . . . . . . . . . 10 ((1st𝑥)𝐺(2nd𝑥) ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
1812, 17bitrdi 287 . . . . . . . . 9 (𝑥𝐺 → (𝑥𝐺 ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)))
1918ibi 267 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
2016snid 4661 . . . . . . . . . . . 12 (2nd𝑥) ∈ {(2nd𝑥)}
21 opelxp 5720 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ ((1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ (2nd𝑥) ∈ {(2nd𝑥)}))
2220, 21mpbiran2 710 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
23 fveq2 6905 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → (𝐹𝑡) = (𝐹‘(2nd𝑥)))
24 sneq 4635 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → {𝑡} = {(2nd𝑥)})
2523, 24xpeq12d 5715 . . . . . . . . . . . . 13 (𝑡 = (2nd𝑥) → ((𝐹𝑡) × {𝑡}) = ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}))
2625eleq2d 2826 . . . . . . . . . . . 12 (𝑡 = (2nd𝑥) → (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})))
2726rspcev 3621 . . . . . . . . . . 11 (((2nd𝑥) ∈ ℕ0 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
2822, 27sylan2br 595 . . . . . . . . . 10 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
29 eliun 4994 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ↔ ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
3028, 29sylibr 234 . . . . . . . . 9 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
31303adant3 1132 . . . . . . . 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 2840 . . . . . 6 (𝑥𝐺𝑥 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3433ssriv 3986 . . . . 5 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
35 ssdomg 9041 . . . . 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 14021 . . . . . . 7 0 ≈ ℕ
38 nnenom 14022 . . . . . . 7 ℕ ≈ ω
3937, 38entri 9049 . . . . . 6 0 ≈ ω
40 endom 9020 . . . . . 6 (ℕ0 ≈ ω → ℕ0 ≼ ω)
4139, 40ax-mp 5 . . . . 5 0 ≼ ω
42 vex 3483 . . . . . . . 8 𝑡 ∈ V
432, 42xpsnen 9096 . . . . . . 7 ((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡)
44 inss2 4237 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
45 heibor.7 . . . . . . . . . 10 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
4645ffvelcdmda 7103 . . . . . . . . 9 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ (𝒫 𝑋 ∩ Fin))
4744, 46sselid 3980 . . . . . . . 8 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ Fin)
48 isfinite 9693 . . . . . . . . 9 ((𝐹𝑡) ∈ Fin ↔ (𝐹𝑡) ≺ ω)
49 sdomdom 9021 . . . . . . . . 9 ((𝐹𝑡) ≺ ω → (𝐹𝑡) ≼ ω)
5048, 49sylbi 217 . . . . . . . 8 ((𝐹𝑡) ∈ Fin → (𝐹𝑡) ≼ ω)
5147, 50syl 17 . . . . . . 7 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ≼ ω)
52 endomtr 9053 . . . . . . 7 ((((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡) ∧ (𝐹𝑡) ≼ ω) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5343, 51, 52sylancr 587 . . . . . 6 ((𝜑𝑡 ∈ ℕ0) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5453ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
55 iunctb 10615 . . . . 5 ((ℕ0 ≼ ω ∧ ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
5641, 54, 55sylancr 587 . . . 4 (𝜑 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
57 domtr 9048 . . . 4 ((𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∧ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝐺 ≼ ω)
5836, 56, 57sylancr 587 . . 3 (𝜑𝐺 ≼ ω)
5919simp1d 1142 . . . . . . . . 9 (𝑥𝐺 → (2nd𝑥) ∈ ℕ0)
60 peano2nn0 12568 . . . . . . . . 9 ((2nd𝑥) ∈ ℕ0 → ((2nd𝑥) + 1) ∈ ℕ0)
6159, 60syl 17 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) + 1) ∈ ℕ0)
62 ffvelcdm 7100 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6345, 61, 62syl2an 596 . . . . . . 7 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6444, 63sselid 3980 . . . . . 6 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ Fin)
65 iunin2 5070 . . . . . . . 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 7439 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
6867cbviunv 5039 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛)
69 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑛 = ((2nd𝑥) + 1) → (𝐹𝑛) = (𝐹‘((2nd𝑥) + 1)))
7069iuneq1d 5018 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
7168, 70eqtrid 2788 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
72 oveq2 7440 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd𝑥) + 1)))
7372iuneq2d 5021 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7471, 73eqtrd 2776 . . . . . . . . . . . . 13 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7574eqeq2d 2747 . . . . . . . . . . . 12 (𝑛 = ((2nd𝑥) + 1) → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
7675rspccva 3620 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7766, 61, 76syl2an 596 . . . . . . . . . 10 ((𝜑𝑥𝐺) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7877ineq2d 4219 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
799fveq2d 6909 . . . . . . . . . . . . . 14 (𝑥𝐺 → (𝐵𝑥) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩))
80 df-ov 7435 . . . . . . . . . . . . . 14 ((1st𝑥)𝐵(2nd𝑥)) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩)
8179, 80eqtr4di 2794 . . . . . . . . . . . . 13 (𝑥𝐺 → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
8281adantl 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
83 inss1 4236 . . . . . . . . . . . . . . . 16 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84 ffvelcdm 7100 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (2nd𝑥) ∈ ℕ0) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8545, 59, 84syl2an 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8683, 85sselid 3980 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ 𝒫 𝑋)
8786elpwid 4608 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ⊆ 𝑋)
8819simp2d 1143 . . . . . . . . . . . . . . 15 (𝑥𝐺 → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
8988adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
9087, 89sseldd 3983 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1st𝑥) ∈ 𝑋)
9159adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (2nd𝑥) ∈ ℕ0)
92 oveq1 7439 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93 oveq2 7440 . . . . . . . . . . . . . . . 16 (𝑚 = (2nd𝑥) → (2↑𝑚) = (2↑(2nd𝑥)))
9493oveq2d 7448 . . . . . . . . . . . . . . 15 (𝑚 = (2nd𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd𝑥))))
9594oveq2d 7448 . . . . . . . . . . . . . 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 7465 . . . . . . . . . . . . . 14 ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ∈ V
9892, 95, 96, 97ovmpo 7594 . . . . . . . . . . . . 13 (((1st𝑥) ∈ 𝑋 ∧ (2nd𝑥) ∈ ℕ0) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
9990, 91, 98syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
10082, 99eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
101 heibor.6 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (CMet‘𝑋))
102 cmetmet 25321 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103101, 102syl 17 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ (Met‘𝑋))
104 metxmet 24345 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105103, 104syl 17 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (∞Met‘𝑋))
106105adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107 2nn 12340 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
108 nnexpcl 14116 . . . . . . . . . . . . . . . 16 ((2 ∈ ℕ ∧ (2nd𝑥) ∈ ℕ0) → (2↑(2nd𝑥)) ∈ ℕ)
109107, 91, 108sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℕ)
110109nnrpd 13076 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℝ+)
111110rpreccld 13088 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ+)
112111rpxrd 13079 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ*)
113 blssm 24429 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑥) ∈ 𝑋 ∧ (1 / (2↑(2nd𝑥))) ∈ ℝ*) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
114106, 90, 112, 113syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
115100, 114eqsstrd 4017 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑋)
116 dfss2 3968 . . . . . . . . . 10 ((𝐵𝑥) ⊆ 𝑋 ↔ ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
117115, 116sylib 218 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
11878, 117eqtr3d 2778 . . . . . . . 8 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
11965, 118eqtrid 2788 . . . . . . 7 ((𝜑𝑥𝐺) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
120 eqimss2 4042 . . . . . . 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 1144 . . . . . . . 8 (𝑥𝐺 → ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)
12381, 122eqeltrd 2840 . . . . . . 7 (𝑥𝐺 → (𝐵𝑥) ∈ 𝐾)
124123adantl 481 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ∈ 𝐾)
125 fvex 6918 . . . . . . . 8 (𝐵𝑥) ∈ V
126125inex1 5316 . . . . . . 7 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ V
12713, 14, 126heiborlem1 37819 . . . . . 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 1372 . . . . 5 ((𝜑𝑥𝐺) → ∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
12983, 63sselid 3980 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ 𝒫 𝑋)
130129elpwid 4608 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝑋)
13113mopnuni 24452 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
132105, 131syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = 𝐽)
133132adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → 𝑋 = 𝐽)
134130, 133sseqtrd 4019 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝐽)
135134sselda 3982 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))) → 𝑡 𝐽)
136135adantrr 717 . . . . . . . 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 9084 . . . . . . . . . . . 12 {(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin
140 inss2 4237 . . . . . . . . . . . . 13 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ (𝑡𝐵((2nd𝑥) + 1))
141 ovex 7465 . . . . . . . . . . . . . . 15 (𝑡𝐵((2nd𝑥) + 1)) ∈ V
142141unisn 4925 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = (𝑡𝐵((2nd𝑥) + 1))
143 uniiun 5057 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
144142, 143eqtr3i 2766 . . . . . . . . . . . . 13 (𝑡𝐵((2nd𝑥) + 1)) = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
145140, 144sseqtri 4031 . . . . . . . . . . . 12 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
146 vex 3483 . . . . . . . . . . . . 13 𝑔 ∈ V
14713, 14, 146heiborlem1 37819 . . . . . . . . . . . 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 1452 . . . . . . . . . . 11 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → ∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾)
149 eleq1 2828 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐵((2nd𝑥) + 1)) → (𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
150141, 149rexsn 4681 . . . . . . . . . . 11 (∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
151148, 150sylib 218 . . . . . . . . . 10 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
152 ovex 7465 . . . . . . . . . . . 12 ((2nd𝑥) + 1) ∈ V
15313, 14, 6, 42, 152heiborlem2 37820 . . . . . . . . . . 11 (𝑡𝐺((2nd𝑥) + 1) ↔ (((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
154153biimpri 228 . . . . . . . . . 10 ((((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
155137, 138, 151, 154syl3an 1160 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
1561553expb 1120 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡𝐺((2nd𝑥) + 1))
157 simprr 772 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
158136, 156, 157jca32 515 . . . . . . 7 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
159158ex 412 . . . . . 6 ((𝜑𝑥𝐺) → ((𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))))
160159reximdv2 3163 . . . . 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 3145 . . 3 (𝜑 → ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
16313fvexi 6919 . . . . 5 𝐽 ∈ V
164163uniex 7762 . . . 4 𝐽 ∈ V
165 breq1 5145 . . . . 5 (𝑡 = (𝑔𝑥) → (𝑡𝐺((2nd𝑥) + 1) ↔ (𝑔𝑥)𝐺((2nd𝑥) + 1)))
166 oveq1 7439 . . . . . . 7 (𝑡 = (𝑔𝑥) → (𝑡𝐵((2nd𝑥) + 1)) = ((𝑔𝑥)𝐵((2nd𝑥) + 1)))
167166ineq2d 4219 . . . . . 6 (𝑡 = (𝑔𝑥) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))))
168167eleq1d 2825 . . . . 5 (𝑡 = (𝑔𝑥) → (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
169165, 168anbi12d 632 . . . 4 (𝑡 = (𝑔𝑥) → ((𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
170164, 169axcc4dom 10482 . . 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)))
17158, 162, 170syl2anc 584 . 2 (𝜑 → ∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
172 exsimpr 1868 . 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))
173171, 172syl 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 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  𝒫 cpw 4599  {csn 4625  cop 4631   cuni 4906   ciun 4990   class class class wbr 5142  {copab 5204   × cxp 5682  Rel wrel 5689  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  ωcom 7888  1st c1st 8013  2nd c2nd 8014  cen 8983  cdom 8984  csdm 8985  Fincfn 8986  1c1 11157   + caddc 11159  *cxr 11295   / cdiv 11921  cn 12267  2c2 12322  0cn0 12528  cexp 14103  ∞Metcxmet 21350  Metcmet 21351  ballcbl 21352  MetOpencmopn 21355  CMetccmet 25289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cc 10476  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-acn 9983  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-seq 14044  df-exp 14104  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954  df-cmet 25292
This theorem is referenced by:  heiborlem10  37828
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