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Theorem heiborlem3 35196
Description: Lemma for heibor 35204. Using countable choice ax-cc 9855, 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 35194 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 9855 via iunctb 9994), and so we can apply ax-cc 9855 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 11900 . . . . . 6 0 ∈ V
2 fvex 6674 . . . . . . 7 (𝐹𝑡) ∈ V
3 snex 5319 . . . . . . 7 {𝑡} ∈ V
42, 3xpex 7470 . . . . . 6 ((𝐹𝑡) × {𝑡}) ∈ V
51, 4iunex 7664 . . . . 5 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∈ V
6 heibor.4 . . . . . . . . 9 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
76relopabi 5681 . . . . . . . 8 Rel 𝐺
8 1st2nd 7733 . . . . . . . 8 ((Rel 𝐺𝑥𝐺) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
97, 8mpan 689 . . . . . . 7 (𝑥𝐺𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
109eleq1d 2900 . . . . . . . . . . 11 (𝑥𝐺 → (𝑥𝐺 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺))
11 df-br 5053 . . . . . . . . . . 11 ((1st𝑥)𝐺(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐺)
1210, 11syl6bbr 292 . . . . . . . . . 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 6674 . . . . . . . . . . 11 (1st𝑥) ∈ V
16 fvex 6674 . . . . . . . . . . 11 (2nd𝑥) ∈ V
1713, 14, 6, 15, 16heiborlem2 35195 . . . . . . . . . 10 ((1st𝑥)𝐺(2nd𝑥) ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
1812, 17syl6bb 290 . . . . . . . . 9 (𝑥𝐺 → (𝑥𝐺 ↔ ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾)))
1918ibi 270 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ ((1st𝑥)𝐵(2nd𝑥)) ∈ 𝐾))
2016snid 4586 . . . . . . . . . . . 12 (2nd𝑥) ∈ {(2nd𝑥)}
21 opelxp 5578 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ ((1st𝑥) ∈ (𝐹‘(2nd𝑥)) ∧ (2nd𝑥) ∈ {(2nd𝑥)}))
2220, 21mpbiran2 709 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}) ↔ (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
23 fveq2 6661 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → (𝐹𝑡) = (𝐹‘(2nd𝑥)))
24 sneq 4560 . . . . . . . . . . . . . 14 (𝑡 = (2nd𝑥) → {𝑡} = {(2nd𝑥)})
2523, 24xpeq12d 5573 . . . . . . . . . . . . 13 (𝑡 = (2nd𝑥) → ((𝐹𝑡) × {𝑡}) = ((𝐹‘(2nd𝑥)) × {(2nd𝑥)}))
2625eleq2d 2901 . . . . . . . . . . . 12 (𝑡 = (2nd𝑥) → (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})))
2726rspcev 3609 . . . . . . . . . . 11 (((2nd𝑥) ∈ ℕ0 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹‘(2nd𝑥)) × {(2nd𝑥)})) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
2822, 27sylan2br 597 . . . . . . . . . 10 (((2nd𝑥) ∈ ℕ0 ∧ (1st𝑥) ∈ (𝐹‘(2nd𝑥))) → ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
29 eliun 4909 . . . . . . . . . 10 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ↔ ∃𝑡 ∈ ℕ0 ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐹𝑡) × {𝑡}))
3028, 29sylibr 237 . . . . . . . . 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 2916 . . . . . 6 (𝑥𝐺𝑥 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}))
3433ssriv 3957 . . . . 5 𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡})
35 ssdomg 8551 . . . . 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 13351 . . . . . . 7 0 ≈ ℕ
38 nnenom 13352 . . . . . . 7 ℕ ≈ ω
3937, 38entri 8559 . . . . . 6 0 ≈ ω
40 endom 8532 . . . . . 6 (ℕ0 ≈ ω → ℕ0 ≼ ω)
4139, 40ax-mp 5 . . . . 5 0 ≼ ω
42 vex 3483 . . . . . . . 8 𝑡 ∈ V
432, 42xpsnen 8597 . . . . . . 7 ((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡)
44 inss2 4191 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
45 heibor.7 . . . . . . . . . 10 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
4645ffvelrnda 6842 . . . . . . . . 9 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ (𝒫 𝑋 ∩ Fin))
4744, 46sseldi 3951 . . . . . . . 8 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ∈ Fin)
48 isfinite 9112 . . . . . . . . 9 ((𝐹𝑡) ∈ Fin ↔ (𝐹𝑡) ≺ ω)
49 sdomdom 8533 . . . . . . . . 9 ((𝐹𝑡) ≺ ω → (𝐹𝑡) ≼ ω)
5048, 49sylbi 220 . . . . . . . 8 ((𝐹𝑡) ∈ Fin → (𝐹𝑡) ≼ ω)
5147, 50syl 17 . . . . . . 7 ((𝜑𝑡 ∈ ℕ0) → (𝐹𝑡) ≼ ω)
52 endomtr 8563 . . . . . . 7 ((((𝐹𝑡) × {𝑡}) ≈ (𝐹𝑡) ∧ (𝐹𝑡) ≼ ω) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5343, 51, 52sylancr 590 . . . . . 6 ((𝜑𝑡 ∈ ℕ0) → ((𝐹𝑡) × {𝑡}) ≼ ω)
5453ralrimiva 3177 . . . . 5 (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
55 iunctb 9994 . . . . 5 ((ℕ0 ≼ ω ∧ ∀𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
5641, 54, 55sylancr 590 . . . 4 (𝜑 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω)
57 domtr 8558 . . . 4 ((𝐺 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ∧ 𝑡 ∈ ℕ0 ((𝐹𝑡) × {𝑡}) ≼ ω) → 𝐺 ≼ ω)
5836, 56, 57sylancr 590 . . 3 (𝜑𝐺 ≼ ω)
5919simp1d 1139 . . . . . . . . 9 (𝑥𝐺 → (2nd𝑥) ∈ ℕ0)
60 peano2nn0 11934 . . . . . . . . 9 ((2nd𝑥) ∈ ℕ0 → ((2nd𝑥) + 1) ∈ ℕ0)
6159, 60syl 17 . . . . . . . 8 (𝑥𝐺 → ((2nd𝑥) + 1) ∈ ℕ0)
62 ffvelrn 6840 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6345, 61, 62syl2an 598 . . . . . . 7 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
6444, 63sseldi 3951 . . . . . 6 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ Fin)
65 iunin2 4979 . . . . . . . 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 7156 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
6867cbviunv 4951 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛)
69 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑛 = ((2nd𝑥) + 1) → (𝐹𝑛) = (𝐹‘((2nd𝑥) + 1)))
7069iuneq1d 4932 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹𝑛)(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
7168, 70syl5eq 2871 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛))
72 oveq2 7157 . . . . . . . . . . . . . . 15 (𝑛 = ((2nd𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd𝑥) + 1)))
7372iuneq2d 4934 . . . . . . . . . . . . . 14 (𝑛 = ((2nd𝑥) + 1) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7471, 73eqtrd 2859 . . . . . . . . . . . . 13 (𝑛 = ((2nd𝑥) + 1) → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7574eqeq2d 2835 . . . . . . . . . . . 12 (𝑛 = ((2nd𝑥) + 1) → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
7675rspccva 3608 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ ((2nd𝑥) + 1) ∈ ℕ0) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7766, 61, 76syl2an 598 . . . . . . . . . 10 ((𝜑𝑥𝐺) → 𝑋 = 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1)))
7877ineq2d 4174 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))))
799fveq2d 6665 . . . . . . . . . . . . . 14 (𝑥𝐺 → (𝐵𝑥) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩))
80 df-ov 7152 . . . . . . . . . . . . . 14 ((1st𝑥)𝐵(2nd𝑥)) = (𝐵‘⟨(1st𝑥), (2nd𝑥)⟩)
8179, 80syl6eqr 2877 . . . . . . . . . . . . 13 (𝑥𝐺 → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
8281adantl 485 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)𝐵(2nd𝑥)))
83 inss1 4190 . . . . . . . . . . . . . . . 16 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84 ffvelrn 6840 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (2nd𝑥) ∈ ℕ0) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8545, 59, 84syl2an 598 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
8683, 85sseldi 3951 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ∈ 𝒫 𝑋)
8786elpwid 4533 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (𝐹‘(2nd𝑥)) ⊆ 𝑋)
8819simp2d 1140 . . . . . . . . . . . . . . 15 (𝑥𝐺 → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
8988adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (1st𝑥) ∈ (𝐹‘(2nd𝑥)))
9087, 89sseldd 3954 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1st𝑥) ∈ 𝑋)
9159adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (2nd𝑥) ∈ ℕ0)
92 oveq1 7156 . . . . . . . . . . . . . 14 (𝑧 = (1st𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93 oveq2 7157 . . . . . . . . . . . . . . . 16 (𝑚 = (2nd𝑥) → (2↑𝑚) = (2↑(2nd𝑥)))
9493oveq2d 7165 . . . . . . . . . . . . . . 15 (𝑚 = (2nd𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd𝑥))))
9594oveq2d 7165 . . . . . . . . . . . . . 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 7182 . . . . . . . . . . . . . 14 ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))) ∈ V
9892, 95, 96, 97ovmpo 7303 . . . . . . . . . . . . 13 (((1st𝑥) ∈ 𝑋 ∧ (2nd𝑥) ∈ ℕ0) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
9990, 91, 98syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → ((1st𝑥)𝐵(2nd𝑥)) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
10082, 99eqtrd 2859 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐵𝑥) = ((1st𝑥)(ball‘𝐷)(1 / (2↑(2nd𝑥)))))
101 heibor.6 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (CMet‘𝑋))
102 cmetmet 23893 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103101, 102syl 17 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ (Met‘𝑋))
104 metxmet 22944 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105103, 104syl 17 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (∞Met‘𝑋))
106105adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107 2nn 11707 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
108 nnexpcl 13447 . . . . . . . . . . . . . . . 16 ((2 ∈ ℕ ∧ (2nd𝑥) ∈ ℕ0) → (2↑(2nd𝑥)) ∈ ℕ)
109107, 91, 108sylancr 590 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℕ)
110109nnrpd 12426 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐺) → (2↑(2nd𝑥)) ∈ ℝ+)
111110rpreccld 12438 . . . . . . . . . . . . 13 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ+)
112111rpxrd 12429 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (1 / (2↑(2nd𝑥))) ∈ ℝ*)
113 blssm 23028 . . . . . . . . . . . 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 3991 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐵𝑥) ⊆ 𝑋)
116 df-ss 3936 . . . . . . . . . 10 ((𝐵𝑥) ⊆ 𝑋 ↔ ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
117115, 116sylib 221 . . . . . . . . 9 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑋) = (𝐵𝑥))
11878, 117eqtr3d 2861 . . . . . . . 8 ((𝜑𝑥𝐺) → ((𝐵𝑥) ∩ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))(𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
11965, 118syl5eq 2871 . . . . . . 7 ((𝜑𝑥𝐺) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = (𝐵𝑥))
120 eqimss2 4010 . . . . . . 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 2916 . . . . . . 7 (𝑥𝐺 → (𝐵𝑥) ∈ 𝐾)
124123adantl 485 . . . . . 6 ((𝜑𝑥𝐺) → (𝐵𝑥) ∈ 𝐾)
125 fvex 6674 . . . . . . . 8 (𝐵𝑥) ∈ V
126125inex1 5207 . . . . . . 7 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ V
12713, 14, 126heiborlem1 35194 . . . . . 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, 63sseldi 3951 . . . . . . . . . . . 12 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ∈ 𝒫 𝑋)
130129elpwid 4533 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝑋)
13113mopnuni 23051 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
132105, 131syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = 𝐽)
133132adantr 484 . . . . . . . . . . 11 ((𝜑𝑥𝐺) → 𝑋 = 𝐽)
134130, 133sseqtrd 3993 . . . . . . . . . 10 ((𝜑𝑥𝐺) → (𝐹‘((2nd𝑥) + 1)) ⊆ 𝐽)
135134sselda 3953 . . . . . . . . 9 (((𝜑𝑥𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd𝑥) + 1))) → 𝑡 𝐽)
136135adantrr 716 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → 𝑡 𝐽)
13761adantl 485 . . . . . . . . . 10 ((𝜑𝑥𝐺) → ((2nd𝑥) + 1) ∈ ℕ0)
138 id 22 . . . . . . . . . 10 (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) → 𝑡 ∈ (𝐹‘((2nd𝑥) + 1)))
139 snfi 8590 . . . . . . . . . . . 12 {(𝑡𝐵((2nd𝑥) + 1))} ∈ Fin
140 inss2 4191 . . . . . . . . . . . . 13 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ (𝑡𝐵((2nd𝑥) + 1))
141 ovex 7182 . . . . . . . . . . . . . . 15 (𝑡𝐵((2nd𝑥) + 1)) ∈ V
142141unisn 4844 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = (𝑡𝐵((2nd𝑥) + 1))
143 uniiun 4968 . . . . . . . . . . . . . 14 {(𝑡𝐵((2nd𝑥) + 1))} = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
144142, 143eqtr3i 2849 . . . . . . . . . . . . 13 (𝑡𝐵((2nd𝑥) + 1)) = 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
145140, 144sseqtri 3989 . . . . . . . . . . . 12 ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ⊆ 𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔
146 vex 3483 . . . . . . . . . . . . 13 𝑔 ∈ V
14713, 14, 146heiborlem1 35194 . . . . . . . . . . . 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 2903 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐵((2nd𝑥) + 1)) → (𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
150141, 149rexsn 4605 . . . . . . . . . . 11 (∃𝑔 ∈ {(𝑡𝐵((2nd𝑥) + 1))}𝑔𝐾 ↔ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
151148, 150sylib 221 . . . . . . . . . 10 (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 → (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾)
152 ovex 7182 . . . . . . . . . . . 12 ((2nd𝑥) + 1) ∈ V
15313, 14, 6, 42, 152heiborlem2 35195 . . . . . . . . . . 11 (𝑡𝐺((2nd𝑥) + 1) ↔ (((2nd𝑥) + 1) ∈ ℕ0𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ (𝑡𝐵((2nd𝑥) + 1)) ∈ 𝐾))
154153biimpri 231 . . . . . . . . . 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 772 . . . . . . . 8 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)
158136, 156, 157jca32 519 . . . . . . 7 (((𝜑𝑥𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
159158ex 416 . . . . . 6 ((𝜑𝑥𝐺) → ((𝑡 ∈ (𝐹‘((2nd𝑥) + 1)) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) → (𝑡 𝐽 ∧ (𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))))
160159reximdv2 3263 . . . . 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 3177 . . 3 (𝜑 → ∀𝑥𝐺𝑡 𝐽(𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾))
16313fvexi 6675 . . . . 5 𝐽 ∈ V
164163uniex 7461 . . . 4 𝐽 ∈ V
165 breq1 5055 . . . . 5 (𝑡 = (𝑔𝑥) → (𝑡𝐺((2nd𝑥) + 1) ↔ (𝑔𝑥)𝐺((2nd𝑥) + 1)))
166 oveq1 7156 . . . . . . 7 (𝑡 = (𝑔𝑥) → (𝑡𝐵((2nd𝑥) + 1)) = ((𝑔𝑥)𝐵((2nd𝑥) + 1)))
167166ineq2d 4174 . . . . . 6 (𝑡 = (𝑔𝑥) → ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) = ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))))
168167eleq1d 2900 . . . . 5 (𝑡 = (𝑔𝑥) → (((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
169165, 168anbi12d 633 . . . 4 (𝑡 = (𝑔𝑥) → ((𝑡𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ (𝑡𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
170164, 169axcc4dom 9861 . . 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 587 . 2 (𝜑 → ∃𝑔(𝑔:𝐺 𝐽 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾)))
172 exsimpr 1871 . 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 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wrex 3134  Vcvv 3480  cin 3918  wss 3919  𝒫 cpw 4522  {csn 4550  cop 4556   cuni 4824   ciun 4905   class class class wbr 5052  {copab 5114   × cxp 5540  Rel wrel 5547  wf 6339  cfv 6343  (class class class)co 7149  cmpo 7151  ωcom 7574  1st c1st 7682  2nd c2nd 7683  cen 8502  cdom 8503  csdm 8504  Fincfn 8505  1c1 10536   + caddc 10538  *cxr 10672   / cdiv 11295  cn 11634  2c2 11689  0cn0 11894  cexp 13434  ∞Metcxmet 20530  Metcmet 20531  ballcbl 20532  MetOpencmopn 20535  CMetccmet 23861
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cc 9855  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-acn 9368  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-seq 13374  df-exp 13435  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-bases 21554  df-cmet 23864
This theorem is referenced by:  heiborlem10  35203
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