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Theorem heiborlem3 37516
Description: Lemma for heibor 37524. Using countable choice ax-cc 10480, 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 37514 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 10480 via iunctb 10619), and so we can apply ax-cc 10480 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 12532 . . . . . 6 0 ∈ V
2 fvex 6916 . . . . . . 7 (𝐹𝑡) ∈ V
3 vsnex 5437 . . . . . . 7 {𝑡} ∈ V
42, 3xpex 7763 . . . . . 6 ((𝐹𝑡) × {𝑡}) ∈ V
51, 4iunex 7984 . . . . 5 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V
6 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
76relopabiv 5828 . . . . . . . 8 Rel 𝐺
8 1st2nd 8055 . . . . . . . 8 ((Rel 𝐺𝑥𝐺) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
97, 8mpan 688 . . . . . . 7 (𝑥𝐺𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109eleq1d 2811 . . . . . . . . . . 11 (𝑥𝐺 → (𝑥𝐺 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺))
11 df-br 5156 . . . . . . . . . . 11 ((1st𝑥)𝐺(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺)
1210, 11bitr4di 288 . . . . . . . . . 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 6916 . . . . . . . . . . 11 (1st𝑥) ∈ V
16 fvex 6916 . . . . . . . . . . 11 (2nd𝑥) ∈ V
1713, 14, 6, 15, 16heiborlem2 37515 . . . . . . . . . 10 ((1st𝑥)𝐺(2nd𝑥) ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
1812, 17bitrdi 286 . . . . . . . . 9 (𝑥𝐺 → (𝑥𝐺 ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)))
1918ibi 266 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
2016snid 4669 . . . . . . . . . . . 12 (2nd𝑥) ∈ {(2nd𝑥)}
21 opelxp 5720 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ ((1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ (2nd𝑥) ∈ {(2nd𝑥)}))
2220, 21mpbiran2 708 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
23 fveq2 6903 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → (𝐹𝑡) = (𝐹‘(2nd𝑥)))
24 sneq 4643 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → {𝑡} = {(2nd𝑥)})
2523, 24xpeq12d 5715 . . . . . . . . . . . . 13 (𝑡 = (2nd𝑥) → ((𝐹𝑡) × {𝑡}) = ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}))
2625eleq2d 2812 . . . . . . . . . . . 12 (𝑡 = (2nd𝑥) → (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})))
2726rspcev 3608 . . . . . . . . . . 11 (((2nd𝑥) ∈ ℕ0 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
2822, 27sylan2br 593 . . . . . . . . . 10 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
29 eliun 5007 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ↔ ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
3028, 29sylibr 233 . . . . . . . . 9 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
31303adant3 1129 . . . . . . . 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 2826 . . . . . 6 (𝑥𝐺𝑥 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3433ssriv 3983 . . . . 5 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
35 ssdomg 9033 . . . . 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 14001 . . . . . . 7 0 ≈ ℕ
38 nnenom 14002 . . . . . . 7 ℕ ≈ ω
3937, 38entri 9041 . . . . . 6 0 ≈ ω
40 endom 9012 . . . . . 6 (ℕ0 ≈ ω → ℕ0 ≼ ω)
4139, 40ax-mp 5 . . . . 5 0 ≼ ω
42 vex 3466 . . . . . . . 8 𝑡 ∈ V
432, 42xpsnen 9095 . . . . . . 7 ((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡)
44 inss2 4231 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
45 heibor.7 . . . . . . . . . 10 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
4645ffvelcdmda 7100 . . . . . . . . 9 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ (𝒫 𝑋 ∩ Fin))
4744, 46sselid 3977 . . . . . . . 8 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ Fin)
48 isfinite 9697 . . . . . . . . 9 ((𝐹𝑡) ∈ Fin ↔ (𝐹𝑡) ≺ ω)
49 sdomdom 9013 . . . . . . . . 9 ((𝐹𝑡) ≺ ω → (𝐹𝑡) ≼ ω)
5048, 49sylbi 216 . . . . . . . 8 ((𝐹𝑡) ∈ Fin → (𝐹𝑡) ≼ ω)
5147, 50syl 17 . . . . . . 7 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ≼ ω)
52 endomtr 9045 . . . . . . 7 ((((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡) ∧ (𝐹𝑡) ≼ ω) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5343, 51, 52sylancr 585 . . . . . 6 ((𝜑𝑡 ∈ ℕ0) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5453ralrimiva 3136 . . . . 5 (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
55 iunctb 10619 . . . . 5 ((ℕ0 ≼ ω ∧ ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
5641, 54, 55sylancr 585 . . . 4 (𝜑 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
57 domtr 9040 . . . 4 ((𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∧ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝐺 ≼ ω)
5836, 56, 57sylancr 585 . . 3 (𝜑𝐺 ≼ ω)
5919simp1d 1139 . . . . . . . . 9 (𝑥𝐺 → (2nd𝑥) ∈ ℕ0)
60 peano2nn0 12566 . . . . . . . . 9 ((2nd𝑥) ∈ ℕ0 → ((2nd𝑥) + 1) ∈ ℕ0)
6159, 60syl 17 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) + 1) ∈ ℕ0)
62 ffvelcdm 7097 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6345, 61, 62syl2an 594 . . . . . . 7 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6444, 63sselid 3977 . . . . . 6 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ Fin)
65 iunin2 5081 . . . . . . . 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 7433 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
6867cbviunv 5050 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛)
69 fveq2 6903 . . . . . . . . . . . . . . . 16 (𝑛 = ((2nd𝑥) + 1) → (𝐹𝑛) = (𝐹‘((2nd𝑥) + 1)))
7069iuneq1d 5030 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
7168, 70eqtrid 2778 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
72 oveq2 7434 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd𝑥) + 1)))
7372iuneq2d 5032 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7471, 73eqtrd 2766 . . . . . . . . . . . . 13 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7574eqeq2d 2737 . . . . . . . . . . . 12 (𝑛 = ((2nd𝑥) + 1) → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
7675rspccva 3607 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7766, 61, 76syl2an 594 . . . . . . . . . 10 ((𝜑𝑥𝐺) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7877ineq2d 4213 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
799fveq2d 6907 . . . . . . . . . . . . . 14 (𝑥𝐺 → (𝐵𝑥) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩))
80 df-ov 7429 . . . . . . . . . . . . . 14 ((1st𝑥)𝐵(2nd𝑥)) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩)
8179, 80eqtr4di 2784 . . . . . . . . . . . . 13 (𝑥𝐺 → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
8281adantl 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
83 inss1 4230 . . . . . . . . . . . . . . . 16 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84 ffvelcdm 7097 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (2nd𝑥) ∈ ℕ0) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8545, 59, 84syl2an 594 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8683, 85sselid 3977 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ 𝒫 𝑋)
8786elpwid 4616 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ⊆ 𝑋)
8819simp2d 1140 . . . . . . . . . . . . . . 15 (𝑥𝐺 → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
8988adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
9087, 89sseldd 3980 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1st𝑥) ∈ 𝑋)
9159adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (2nd𝑥) ∈ ℕ0)
92 oveq1 7433 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93 oveq2 7434 . . . . . . . . . . . . . . . 16 (𝑚 = (2nd𝑥) → (2↑𝑚) = (2↑(2nd𝑥)))
9493oveq2d 7442 . . . . . . . . . . . . . . 15 (𝑚 = (2nd𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd𝑥))))
9594oveq2d 7442 . . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . 14 ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ∈ V
9892, 95, 96, 97ovmpo 7588 . . . . . . . . . . . . 13 (((1st𝑥) ∈ 𝑋 ∧ (2nd𝑥) ∈ ℕ0) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
9990, 91, 98syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
10082, 99eqtrd 2766 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
101 heibor.6 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (CMet‘𝑋))
102 cmetmet 25308 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103101, 102syl 17 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ (Met‘𝑋))
104 metxmet 24334 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105103, 104syl 17 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (∞Met‘𝑋))
106105adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107 2nn 12339 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
108 nnexpcl 14096 . . . . . . . . . . . . . . . 16 ((2 ∈ ℕ ∧ (2nd𝑥) ∈ ℕ0) → (2↑(2nd𝑥)) ∈ ℕ)
109107, 91, 108sylancr 585 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℕ)
110109nnrpd 13070 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℝ+)
111110rpreccld 13082 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ+)
112111rpxrd 13073 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ*)
113 blssm 24418 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑥) ∈ 𝑋 ∧ (1 / (2↑(2nd𝑥))) ∈ ℝ*) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
114106, 90, 112, 113syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
115100, 114eqsstrd 4018 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑋)
116 dfss2 3965 . . . . . . . . . 10 ((𝐵𝑥) ⊆ 𝑋 ↔ ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
117115, 116sylib 217 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
11878, 117eqtr3d 2768 . . . . . . . 8 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
11965, 118eqtrid 2778 . . . . . . 7 ((𝜑𝑥𝐺) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
120 eqimss2 4039 . . . . . . 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 1141 . . . . . . . 8 (𝑥𝐺 → ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)
12381, 122eqeltrd 2826 . . . . . . 7 (𝑥𝐺 → (𝐵𝑥) ∈ 𝐾)
124123adantl 480 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ∈ 𝐾)
125 fvex 6916 . . . . . . . 8 (𝐵𝑥) ∈ V
126125inex1 5324 . . . . . . 7 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ V
12713, 14, 126heiborlem1 37514 . . . . . 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 1368 . . . . 5 ((𝜑𝑥𝐺) → ∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
12983, 63sselid 3977 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ 𝒫 𝑋)
130129elpwid 4616 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝑋)
13113mopnuni 24441 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
132105, 131syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = 𝐽)
133132adantr 479 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → 𝑋 = 𝐽)
134130, 133sseqtrd 4020 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝐽)
135134sselda 3979 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))) → 𝑡 𝐽)
136135adantrr 715 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡 𝐽)
13761adantl 480 . . . . . . . . . 10 ((𝜑𝑥𝐺) → ((2nd𝑥) + 1) ∈ ℕ0)
138 id 22 . . . . . . . . . 10 (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)))
139 snfi 9083 . . . . . . . . . . . 12 {(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin
140 inss2 4231 . . . . . . . . . . . . 13 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ (𝑡𝐵((2nd𝑥) + 1))
141 ovex 7459 . . . . . . . . . . . . . . 15 (𝑡𝐵((2nd𝑥) + 1)) ∈ V
142141unisn 4936 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = (𝑡𝐵((2nd𝑥) + 1))
143 uniiun 5068 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
144142, 143eqtr3i 2756 . . . . . . . . . . . . 13 (𝑡𝐵((2nd𝑥) + 1)) = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
145140, 144sseqtri 4016 . . . . . . . . . . . 12 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
146 vex 3466 . . . . . . . . . . . . 13 𝑔 ∈ V
14713, 14, 146heiborlem1 37514 . . . . . . . . . . . 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 1448 . . . . . . . . . . 11 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → ∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾)
149 eleq1 2814 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐵((2nd𝑥) + 1)) → (𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
150141, 149rexsn 4691 . . . . . . . . . . 11 (∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
151148, 150sylib 217 . . . . . . . . . 10 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
152 ovex 7459 . . . . . . . . . . . 12 ((2nd𝑥) + 1) ∈ V
15313, 14, 6, 42, 152heiborlem2 37515 . . . . . . . . . . 11 (𝑡𝐺((2nd𝑥) + 1) ↔ (((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
154153biimpri 227 . . . . . . . . . 10 ((((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
155137, 138, 151, 154syl3an 1157 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → 𝑡𝐺((2nd𝑥) + 1))
1561553expb 1117 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡𝐺((2nd𝑥) + 1))
157 simprr 771 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
158136, 156, 157jca32 514 . . . . . . 7 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
159158ex 411 . . . . . 6 ((𝜑𝑥𝐺) → ((𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))))
160159reximdv2 3154 . . . . 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 3136 . . 3 (𝜑 → ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
16313fvexi 6917 . . . . 5 𝐽 ∈ V
164163uniex 7754 . . . 4 𝐽 ∈ V
165 breq1 5158 . . . . 5 (𝑡 = (𝑔𝑥) → (𝑡𝐺((2nd𝑥) + 1) ↔ (𝑔𝑥)𝐺((2nd𝑥) + 1)))
166 oveq1 7433 . . . . . . 7 (𝑡 = (𝑔𝑥) → (𝑡𝐵((2nd𝑥) + 1)) = ((𝑔𝑥)𝐵((2nd𝑥) + 1)))
167166ineq2d 4213 . . . . . 6 (𝑡 = (𝑔𝑥) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))))
168167eleq1d 2811 . . . . 5 (𝑡 = (𝑔𝑥) → (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
169165, 168anbi12d 630 . . . 4 (𝑡 = (𝑔𝑥) → ((𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
170164, 169axcc4dom 10486 . . 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 582 . 2 (𝜑 → ∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
172 exsimpr 1865 . 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 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wral 3051  wrex 3060  Vcvv 3462  cin 3946  wss 3947  𝒫 cpw 4607  {csn 4633  cop 4639   cuni 4915   ciun 5003   class class class wbr 5155  {copab 5217   × cxp 5682  Rel wrel 5689  wf 6552  cfv 6556  (class class class)co 7426  cmpo 7428  ωcom 7878  1st c1st 8003  2nd c2nd 8004  cen 8973  cdom 8974  csdm 8975  Fincfn 8976  1c1 11161   + caddc 11163  *cxr 11299   / cdiv 11923  cn 12266  2c2 12321  0cn0 12526  cexp 14083  ∞Metcxmet 21330  Metcmet 21331  ballcbl 21332  MetOpencmopn 21335  CMetccmet 25276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-inf2 9686  ax-cc 10480  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237  ax-pre-sup 11238
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-se 5640  df-we 5641  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 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-isom 6565  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-er 8736  df-map 8859  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-sup 9487  df-inf 9488  df-oi 9555  df-card 9984  df-acn 9987  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-div 11924  df-nn 12267  df-2 12329  df-n0 12527  df-z 12613  df-uz 12877  df-q 12987  df-rp 13031  df-xneg 13148  df-xadd 13149  df-xmul 13150  df-seq 14024  df-exp 14084  df-topgen 17460  df-psmet 21337  df-xmet 21338  df-met 21339  df-bl 21340  df-mopn 21341  df-top 22890  df-topon 22907  df-bases 22943  df-cmet 25279
This theorem is referenced by:  heiborlem10  37523
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