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Theorem heiborlem3 36272
Description: Lemma for heibor 36280. Using countable choice ax-cc 10371, 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 36270 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 10371 via iunctb 10510), and so we can apply ax-cc 10371 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 12419 . . . . . 6 0 ∈ V
2 fvex 6855 . . . . . . 7 (𝐹𝑡) ∈ V
3 vsnex 5386 . . . . . . 7 {𝑡} ∈ V
42, 3xpex 7687 . . . . . 6 ((𝐹𝑡) × {𝑡}) ∈ V
51, 4iunex 7901 . . . . 5 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V
6 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
76relopabiv 5776 . . . . . . . 8 Rel 𝐺
8 1st2nd 7971 . . . . . . . 8 ((Rel 𝐺𝑥𝐺) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
97, 8mpan 688 . . . . . . 7 (𝑥𝐺𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109eleq1d 2822 . . . . . . . . . . 11 (𝑥𝐺 → (𝑥𝐺 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺))
11 df-br 5106 . . . . . . . . . . 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 6855 . . . . . . . . . . 11 (1st𝑥) ∈ V
16 fvex 6855 . . . . . . . . . . 11 (2nd𝑥) ∈ V
1713, 14, 6, 15, 16heiborlem2 36271 . . . . . . . . . 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 4622 . . . . . . . . . . . 12 (2nd𝑥) ∈ {(2nd𝑥)}
21 opelxp 5669 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → (𝐹𝑡) = (𝐹‘(2nd𝑥)))
24 sneq 4596 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → {𝑡} = {(2nd𝑥)})
2523, 24xpeq12d 5664 . . . . . . . . . . . . 13 (𝑡 = (2nd𝑥) → ((𝐹𝑡) × {𝑡}) = ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}))
2625eleq2d 2823 . . . . . . . . . . . 12 (𝑡 = (2nd𝑥) → (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})))
2726rspcev 3581 . . . . . . . . . . 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 4958 . . . . . . . . . 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 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 2838 . . . . . 6 (𝑥𝐺𝑥 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3433ssriv 3948 . . . . 5 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
35 ssdomg 8940 . . . . 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 13884 . . . . . . 7 0 ≈ ℕ
38 nnenom 13885 . . . . . . 7 ℕ ≈ ω
3937, 38entri 8948 . . . . . 6 0 ≈ ω
40 endom 8919 . . . . . 6 (ℕ0 ≈ ω → ℕ0 ≼ ω)
4139, 40ax-mp 5 . . . . 5 0 ≼ ω
42 vex 3449 . . . . . . . 8 𝑡 ∈ V
432, 42xpsnen 8999 . . . . . . 7 ((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡)
44 inss2 4189 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
45 heibor.7 . . . . . . . . . 10 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
4645ffvelcdmda 7035 . . . . . . . . 9 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ (𝒫 𝑋 ∩ Fin))
4744, 46sselid 3942 . . . . . . . 8 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ Fin)
48 isfinite 9588 . . . . . . . . 9 ((𝐹𝑡) ∈ Fin ↔ (𝐹𝑡) ≺ ω)
49 sdomdom 8920 . . . . . . . . 9 ((𝐹𝑡) ≺ ω → (𝐹𝑡) ≼ ω)
5048, 49sylbi 216 . . . . . . . 8 ((𝐹𝑡) ∈ Fin → (𝐹𝑡) ≼ ω)
5147, 50syl 17 . . . . . . 7 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ≼ ω)
52 endomtr 8952 . . . . . . 7 ((((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡) ∧ (𝐹𝑡) ≼ ω) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5343, 51, 52sylancr 587 . . . . . 6 ((𝜑𝑡 ∈ ℕ0) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5453ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
55 iunctb 10510 . . . . 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 8947 . . . 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 12453 . . . . . . . . 9 ((2nd𝑥) ∈ ℕ0 → ((2nd𝑥) + 1) ∈ ℕ0)
6159, 60syl 17 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) + 1) ∈ ℕ0)
62 ffvelcdm 7032 . . . . . . . 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 3942 . . . . . 6 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ Fin)
65 iunin2 5031 . . . . . . . 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 7364 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
6867cbviunv 5000 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛)
69 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = ((2nd𝑥) + 1) → (𝐹𝑛) = (𝐹‘((2nd𝑥) + 1)))
7069iuneq1d 4981 . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd𝑥) + 1)))
7372iuneq2d 4983 . . . . . . . . . . . . . 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 3580 . . . . . . . . . . 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 4172 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
799fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥𝐺 → (𝐵𝑥) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩))
80 df-ov 7360 . . . . . . . . . . . . . 14 ((1st𝑥)𝐵(2nd𝑥)) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩)
8179, 80eqtr4di 2794 . . . . . . . . . . . . 13 (𝑥𝐺 → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
8281adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
83 inss1 4188 . . . . . . . . . . . . . . . 16 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84 ffvelcdm 7032 . . . . . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ 𝒫 𝑋)
8786elpwid 4569 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ⊆ 𝑋)
8819simp2d 1143 . . . . . . . . . . . . . . 15 (𝑥𝐺 → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
8988adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
9087, 89sseldd 3945 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1st𝑥) ∈ 𝑋)
9159adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (2nd𝑥) ∈ ℕ0)
92 oveq1 7364 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = (2nd𝑥) → (2↑𝑚) = (2↑(2nd𝑥)))
9493oveq2d 7373 . . . . . . . . . . . . . . 15 (𝑚 = (2nd𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd𝑥))))
9594oveq2d 7373 . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . 14 ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ∈ V
9892, 95, 96, 97ovmpo 7515 . . . . . . . . . . . . 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 24650 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103101, 102syl 17 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ (Met‘𝑋))
104 metxmet 23687 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105103, 104syl 17 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (∞Met‘𝑋))
106105adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107 2nn 12226 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
108 nnexpcl 13980 . . . . . . . . . . . . . . . 16 ((2 ∈ ℕ ∧ (2nd𝑥) ∈ ℕ0) → (2↑(2nd𝑥)) ∈ ℕ)
109107, 91, 108sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℕ)
110109nnrpd 12955 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℝ+)
111110rpreccld 12967 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ+)
112111rpxrd 12958 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ*)
113 blssm 23771 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑥) ∈ 𝑋 ∧ (1 / (2↑(2nd𝑥))) ∈ ℝ*) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
114106, 90, 112, 113syl3anc 1371 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ⊆ 𝑋)
115100, 114eqsstrd 3982 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑋)
116 df-ss 3927 . . . . . . . . . 10 ((𝐵𝑥) ⊆ 𝑋 ↔ ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
117115, 116sylib 217 . . . . . . . . 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 4001 . . . . . . 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 2838 . . . . . . 7 (𝑥𝐺 → (𝐵𝑥) ∈ 𝐾)
124123adantl 482 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ∈ 𝐾)
125 fvex 6855 . . . . . . . 8 (𝐵𝑥) ∈ V
126125inex1 5274 . . . . . . 7 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ V
12713, 14, 126heiborlem1 36270 . . . . . 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 1371 . . . . 5 ((𝜑𝑥𝐺) → ∃𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
12983, 63sselid 3942 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ 𝒫 𝑋)
130129elpwid 4569 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝑋)
13113mopnuni 23794 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
132105, 131syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = 𝐽)
133132adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → 𝑋 = 𝐽)
134130, 133sseqtrd 3984 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝐽)
135134sselda 3944 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))) → 𝑡 𝐽)
136135adantrr 715 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡 𝐽)
13761adantl 482 . . . . . . . . . 10 ((𝜑𝑥𝐺) → ((2nd𝑥) + 1) ∈ ℕ0)
138 id 22 . . . . . . . . . 10 (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)))
139 snfi 8988 . . . . . . . . . . . 12 {(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin
140 inss2 4189 . . . . . . . . . . . . 13 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ (𝑡𝐵((2nd𝑥) + 1))
141 ovex 7390 . . . . . . . . . . . . . . 15 (𝑡𝐵((2nd𝑥) + 1)) ∈ V
142141unisn 4887 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = (𝑡𝐵((2nd𝑥) + 1))
143 uniiun 5018 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
144142, 143eqtr3i 2766 . . . . . . . . . . . . 13 (𝑡𝐵((2nd𝑥) + 1)) = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
145140, 144sseqtri 3980 . . . . . . . . . . . 12 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
146 vex 3449 . . . . . . . . . . . . 13 𝑔 ∈ V
14713, 14, 146heiborlem1 36270 . . . . . . . . . . . 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 1451 . . . . . . . . . . 11 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → ∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾)
149 eleq1 2825 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐵((2nd𝑥) + 1)) → (𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
150141, 149rexsn 4643 . . . . . . . . . . 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 7390 . . . . . . . . . . . 12 ((2nd𝑥) + 1) ∈ V
15313, 14, 6, 42, 152heiborlem2 36271 . . . . . . . . . . 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 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 771 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
158136, 156, 157jca32 516 . . . . . . 7 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
159158ex 413 . . . . . 6 ((𝜑𝑥𝐺) → ((𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))))
160159reximdv2 3161 . . . . 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 3143 . . 3 (𝜑 → ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
16313fvexi 6856 . . . . 5 𝐽 ∈ V
164163uniex 7678 . . . 4 𝐽 ∈ V
165 breq1 5108 . . . . 5 (𝑡 = (𝑔𝑥) → (𝑡𝐺((2nd𝑥) + 1) ↔ (𝑔𝑥)𝐺((2nd𝑥) + 1)))
166 oveq1 7364 . . . . . . 7 (𝑡 = (𝑔𝑥) → (𝑡𝐵((2nd𝑥) + 1)) = ((𝑔𝑥)𝐵((2nd𝑥) + 1)))
167166ineq2d 4172 . . . . . 6 (𝑡 = (𝑔𝑥) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))))
168167eleq1d 2822 . . . . 5 (𝑡 = (𝑔𝑥) → (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
169165, 168anbi12d 631 . . . 4 (𝑡 = (𝑔𝑥) → ((𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
170164, 169axcc4dom 10377 . . 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 1872 . 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  {copab 5167   × cxp 5631  Rel wrel 5638  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  ωcom 7802  1st c1st 7919  2nd c2nd 7920  cen 8880  cdom 8881  csdm 8882  Fincfn 8883  1c1 11052   + caddc 11054  *cxr 11188   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  cexp 13967  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  CMetccmet 24618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-seq 13907  df-exp 13968  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmet 24621
This theorem is referenced by:  heiborlem10  36279
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