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 (𝐹‘𝑛) 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5	⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
heibor.7	⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))

Assertion

Ref	Expression
heiborlem3	⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 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.

Step	Hyp	Ref	Expression
1		nn0ex 12532	. . . . . 6 ⊢ ℕ0 ∈ V
2		fvex 6916	. . . . . . 7 ⊢ (𝐹‘𝑡) ∈ V
3		vsnex 5437	. . . . . . 7 ⊢ {𝑡} ∈ V
4	2, 3	xpex 7763	. . . . . 6 ⊢ ((𝐹‘𝑡) × {𝑡}) ∈ V
5	1, 4	iunex 7984	. . . . 5 ⊢ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ∈ V
6		heibor.4	. . . . . . . . 9 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
7	6	relopabiv 5828	. . . . . . . 8 ⊢ Rel 𝐺
8		1st2nd 8055	. . . . . . . 8 ⊢ ((Rel 𝐺 ∧ 𝑥 ∈ 𝐺) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
9	7, 8	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ 𝐺 → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
10	9	eleq1d 2811	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐺 → (𝑥 ∈ 𝐺 ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐺))
11		df-br 5156	. . . . . . . . . . 11 ⊢ ((1st ‘𝑥)𝐺(2nd ‘𝑥) ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐺)
12	10, 11	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐺 → (𝑥 ∈ 𝐺 ↔ (1st ‘𝑥)𝐺(2nd ‘𝑥)))
13		heibor.1	. . . . . . . . . . 11 ⊢ 𝐽 = (MetOpen‘𝐷)
14		heibor.3	. . . . . . . . . . 11 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
15		fvex 6916	. . . . . . . . . . 11 ⊢ (1st ‘𝑥) ∈ V
16		fvex 6916	. . . . . . . . . . 11 ⊢ (2nd ‘𝑥) ∈ V
17	13, 14, 6, 15, 16	heiborlem2 37515	. . . . . . . . . 10 ⊢ ((1st ‘𝑥)𝐺(2nd ‘𝑥) ↔ ((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)) ∧ ((1st ‘𝑥)𝐵(2nd ‘𝑥)) ∈ 𝐾))
18	12, 17	bitrdi 286	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐺 → (𝑥 ∈ 𝐺 ↔ ((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)) ∧ ((1st ‘𝑥)𝐵(2nd ‘𝑥)) ∈ 𝐾)))
19	18	ibi 266	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐺 → ((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)) ∧ ((1st ‘𝑥)𝐵(2nd ‘𝑥)) ∈ 𝐾))
20	16	snid 4669	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑥) ∈ {(2nd ‘𝑥)}
21		opelxp 5720	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘(2nd ‘𝑥)) × {(2nd ‘𝑥)}) ↔ ((1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)) ∧ (2nd ‘𝑥) ∈ {(2nd ‘𝑥)}))
22	20, 21	mpbiran2 708	. . . . . . . . . . 11 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘(2nd ‘𝑥)) × {(2nd ‘𝑥)}) ↔ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)))
23		fveq2 6903	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (2nd ‘𝑥) → (𝐹‘𝑡) = (𝐹‘(2nd ‘𝑥)))
24		sneq 4643	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (2nd ‘𝑥) → {𝑡} = {(2nd ‘𝑥)})
25	23, 24	xpeq12d 5715	. . . . . . . . . . . . 13 ⊢ (𝑡 = (2nd ‘𝑥) → ((𝐹‘𝑡) × {𝑡}) = ((𝐹‘(2nd ‘𝑥)) × {(2nd ‘𝑥)}))
26	25	eleq2d 2812	. . . . . . . . . . . 12 ⊢ (𝑡 = (2nd ‘𝑥) → (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘𝑡) × {𝑡}) ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘(2nd ‘𝑥)) × {(2nd ‘𝑥)})))
27	26	rspcev 3608	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥) ∈ ℕ0 ∧ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘(2nd ‘𝑥)) × {(2nd ‘𝑥)})) → ∃𝑡 ∈ ℕ0 ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘𝑡) × {𝑡}))
28	22, 27	sylan2br 593	. . . . . . . . . 10 ⊢ (((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥))) → ∃𝑡 ∈ ℕ0 ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘𝑡) × {𝑡}))
29		eliun 5007	. . . . . . . . . 10 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ↔ ∃𝑡 ∈ ℕ0 ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ((𝐹‘𝑡) × {𝑡}))
30	28, 29	sylibr 233	. . . . . . . . 9 ⊢ (((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥))) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}))
31	30	3adant3 1129	. . . . . . . 8 ⊢ (((2nd ‘𝑥) ∈ ℕ0 ∧ (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)) ∧ ((1st ‘𝑥)𝐵(2nd ‘𝑥)) ∈ 𝐾) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}))
32	19, 31	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐺 → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}))
33	9, 32	eqeltrd 2826	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 → 𝑥 ∈ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}))
34	33	ssriv 3983	. . . . 5 ⊢ 𝐺 ⊆ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡})
35		ssdomg 9033	. . . . 5 ⊢ (∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ∈ V → (𝐺 ⊆ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) → 𝐺 ≼ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡})))
36	5, 34, 35	mp2 9	. . . 4 ⊢ 𝐺 ≼ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡})
37		nn0ennn 14001	. . . . . . 7 ⊢ ℕ0 ≈ ℕ
38		nnenom 14002	. . . . . . 7 ⊢ ℕ ≈ ω
39	37, 38	entri 9041	. . . . . 6 ⊢ ℕ0 ≈ ω
40		endom 9012	. . . . . 6 ⊢ (ℕ0 ≈ ω → ℕ0 ≼ ω)
41	39, 40	ax-mp 5	. . . . 5 ⊢ ℕ0 ≼ ω
42		vex 3466	. . . . . . . 8 ⊢ 𝑡 ∈ V
43	2, 42	xpsnen 9095	. . . . . . 7 ⊢ ((𝐹‘𝑡) × {𝑡}) ≈ (𝐹‘𝑡)
44		inss2 4231	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ Fin
45		heibor.7	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
46	45	ffvelcdmda 7100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℕ0) → (𝐹‘𝑡) ∈ (𝒫 𝑋 ∩ Fin))
47	44, 46	sselid 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ℕ0) → (𝐹‘𝑡) ∈ Fin)
48		isfinite 9697	. . . . . . . . 9 ⊢ ((𝐹‘𝑡) ∈ Fin ↔ (𝐹‘𝑡) ≺ ω)
49		sdomdom 9013	. . . . . . . . 9 ⊢ ((𝐹‘𝑡) ≺ ω → (𝐹‘𝑡) ≼ ω)
50	48, 49	sylbi 216	. . . . . . . 8 ⊢ ((𝐹‘𝑡) ∈ Fin → (𝐹‘𝑡) ≼ ω)
51	47, 50	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ℕ0) → (𝐹‘𝑡) ≼ ω)
52		endomtr 9045	. . . . . . 7 ⊢ ((((𝐹‘𝑡) × {𝑡}) ≈ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≼ ω) → ((𝐹‘𝑡) × {𝑡}) ≼ ω)
53	43, 51, 52	sylancr 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℕ0) → ((𝐹‘𝑡) × {𝑡}) ≼ ω)
54	53	ralrimiva 3136	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ≼ ω)
55		iunctb 10619	. . . . 5 ⊢ ((ℕ0 ≼ ω ∧ ∀𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ≼ ω) → ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ≼ ω)
56	41, 54, 55	sylancr 585	. . . 4 ⊢ (𝜑 → ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ≼ ω)
57		domtr 9040	. . . 4 ⊢ ((𝐺 ≼ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ∧ ∪ 𝑡 ∈ ℕ0 ((𝐹‘𝑡) × {𝑡}) ≼ ω) → 𝐺 ≼ ω)
58	36, 56, 57	sylancr 585	. . 3 ⊢ (𝜑 → 𝐺 ≼ ω)
59	19	simp1d 1139	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐺 → (2nd ‘𝑥) ∈ ℕ0)
60		peano2nn0 12566	. . . . . . . . 9 ⊢ ((2nd ‘𝑥) ∈ ℕ0 → ((2nd ‘𝑥) + 1) ∈ ℕ0)
61	59, 60	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐺 → ((2nd ‘𝑥) + 1) ∈ ℕ0)
62		ffvelcdm 7097	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ ((2nd ‘𝑥) + 1) ∈ ℕ0) → (𝐹‘((2nd ‘𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
63	45, 61, 62	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘((2nd ‘𝑥) + 1)) ∈ (𝒫 𝑋 ∩ Fin))
64	44, 63	sselid 3977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘((2nd ‘𝑥) + 1)) ∈ Fin)
65		iunin2 5081	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑥) ∩ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1)))
66		heibor.8	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
67		oveq1 7433	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦𝐵𝑛) = (𝑡𝐵𝑛))
68	67	cbviunv 5050	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑡 ∈ (𝐹‘𝑛)(𝑡𝐵𝑛)
69		fveq2 6903	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → (𝐹‘𝑛) = (𝐹‘((2nd ‘𝑥) + 1)))
70	69	iuneq1d 5030	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → ∪ 𝑡 ∈ (𝐹‘𝑛)(𝑡𝐵𝑛) = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵𝑛))
71	68, 70	eqtrid 2778	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵𝑛))
72		oveq2 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → (𝑡𝐵𝑛) = (𝑡𝐵((2nd ‘𝑥) + 1)))
73	72	iuneq2d 5032	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵𝑛) = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1)))
74	71, 73	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1)))
75	74	eqeq2d 2737	. . . . . . . . . . . 12 ⊢ (𝑛 = ((2nd ‘𝑥) + 1) → (𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1))))
76	75	rspccva 3607	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ∧ ((2nd ‘𝑥) + 1) ∈ ℕ0) → 𝑋 = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1)))
77	66, 61, 76	syl2an 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑋 = ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1)))
78	77	ineq2d 4213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((𝐵‘𝑥) ∩ 𝑋) = ((𝐵‘𝑥) ∩ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1))))
79	9	fveq2d 6907	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐺 → (𝐵‘𝑥) = (𝐵‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
80		df-ov 7429	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥)𝐵(2nd ‘𝑥)) = (𝐵‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
81	79, 80	eqtr4di 2784	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐺 → (𝐵‘𝑥) = ((1st ‘𝑥)𝐵(2nd ‘𝑥)))
82	81	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐵‘𝑥) = ((1st ‘𝑥)𝐵(2nd ‘𝑥)))
83		inss1 4230	. . . . . . . . . . . . . . . 16 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
84		ffvelcdm 7097	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (2nd ‘𝑥) ∈ ℕ0) → (𝐹‘(2nd ‘𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
85	45, 59, 84	syl2an 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘(2nd ‘𝑥)) ∈ (𝒫 𝑋 ∩ Fin))
86	83, 85	sselid 3977	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘(2nd ‘𝑥)) ∈ 𝒫 𝑋)
87	86	elpwid 4616	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘(2nd ‘𝑥)) ⊆ 𝑋)
88	19	simp2d 1140	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐺 → (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)))
89	88	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (1st ‘𝑥) ∈ (𝐹‘(2nd ‘𝑥)))
90	87, 89	sseldd 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (1st ‘𝑥) ∈ 𝑋)
91	59	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (2nd ‘𝑥) ∈ ℕ0)
92		oveq1 7433	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘𝑥) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑𝑚))))
93		oveq2 7434	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (2nd ‘𝑥) → (2↑𝑚) = (2↑(2nd ‘𝑥)))
94	93	oveq2d 7442	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (2nd ‘𝑥) → (1 / (2↑𝑚)) = (1 / (2↑(2nd ‘𝑥))))
95	94	oveq2d 7442	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (2nd ‘𝑥) → ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))))
96		heibor.5	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
97		ovex 7459	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))) ∈ V
98	92, 95, 96, 97	ovmpo 7588	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) ∈ 𝑋 ∧ (2nd ‘𝑥) ∈ ℕ0) → ((1st ‘𝑥)𝐵(2nd ‘𝑥)) = ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))))
99	90, 91, 98	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((1st ‘𝑥)𝐵(2nd ‘𝑥)) = ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))))
100	82, 99	eqtrd 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐵‘𝑥) = ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))))
101		heibor.6	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
102		cmetmet 25308	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
103	101, 102	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
104		metxmet 24334	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
105	103, 104	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
106	105	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐷 ∈ (∞Met‘𝑋))
107		2nn 12339	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
108		nnexpcl 14096	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ0) → (2↑(2nd ‘𝑥)) ∈ ℕ)
109	107, 91, 108	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (2↑(2nd ‘𝑥)) ∈ ℕ)
110	109	nnrpd 13070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (2↑(2nd ‘𝑥)) ∈ ℝ+)
111	110	rpreccld 13082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (1 / (2↑(2nd ‘𝑥))) ∈ ℝ+)
112	111	rpxrd 13073	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (1 / (2↑(2nd ‘𝑥))) ∈ ℝ*)
113		blssm 24418	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑥) ∈ 𝑋 ∧ (1 / (2↑(2nd ‘𝑥))) ∈ ℝ*) → ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))) ⊆ 𝑋)
114	106, 90, 112, 113	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((1st ‘𝑥)(ball‘𝐷)(1 / (2↑(2nd ‘𝑥)))) ⊆ 𝑋)
115	100, 114	eqsstrd 4018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐵‘𝑥) ⊆ 𝑋)
116		dfss2 3965	. . . . . . . . . 10 ⊢ ((𝐵‘𝑥) ⊆ 𝑋 ↔ ((𝐵‘𝑥) ∩ 𝑋) = (𝐵‘𝑥))
117	115, 116	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((𝐵‘𝑥) ∩ 𝑋) = (𝐵‘𝑥))
118	78, 117	eqtr3d 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((𝐵‘𝑥) ∩ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))(𝑡𝐵((2nd ‘𝑥) + 1))) = (𝐵‘𝑥))
119	65, 118	eqtrid 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) = (𝐵‘𝑥))
120		eqimss2 4039	. . . . . . 7 ⊢ (∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) = (𝐵‘𝑥) → (𝐵‘𝑥) ⊆ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))))
121	119, 120	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐵‘𝑥) ⊆ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))))
122	19	simp3d 1141	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐺 → ((1st ‘𝑥)𝐵(2nd ‘𝑥)) ∈ 𝐾)
123	81, 122	eqeltrd 2826	. . . . . . 7 ⊢ (𝑥 ∈ 𝐺 → (𝐵‘𝑥) ∈ 𝐾)
124	123	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐵‘𝑥) ∈ 𝐾)
125		fvex 6916	. . . . . . . 8 ⊢ (𝐵‘𝑥) ∈ V
126	125	inex1 5324	. . . . . . 7 ⊢ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ V
127	13, 14, 126	heiborlem1 37514	. . . . . 6 ⊢ (((𝐹‘((2nd ‘𝑥) + 1)) ∈ Fin ∧ (𝐵‘𝑥) ⊆ ∪ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∧ (𝐵‘𝑥) ∈ 𝐾) → ∃𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)
128	64, 121, 124, 127	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)
129	83, 63	sselid 3977	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘((2nd ‘𝑥) + 1)) ∈ 𝒫 𝑋)
130	129	elpwid 4616	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘((2nd ‘𝑥) + 1)) ⊆ 𝑋)
131	13	mopnuni 24441	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
132	105, 131	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
133	132	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑋 = ∪ 𝐽)
134	130, 133	sseqtrd 4020	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝐹‘((2nd ‘𝑥) + 1)) ⊆ ∪ 𝐽)
135	134	sselda 3979	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))) → 𝑡 ∈ ∪ 𝐽)
136	135	adantrr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → 𝑡 ∈ ∪ 𝐽)
137	61	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((2nd ‘𝑥) + 1) ∈ ℕ0)
138		id 22	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) → 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)))
139		snfi 9083	. . . . . . . . . . . 12 ⊢ {(𝑡𝐵((2nd ‘𝑥) + 1))} ∈ Fin
140		inss2 4231	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ⊆ (𝑡𝐵((2nd ‘𝑥) + 1))
141		ovex 7459	. . . . . . . . . . . . . . 15 ⊢ (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ V
142	141	unisn 4936	. . . . . . . . . . . . . 14 ⊢ ∪ {(𝑡𝐵((2nd ‘𝑥) + 1))} = (𝑡𝐵((2nd ‘𝑥) + 1))
143		uniiun 5068	. . . . . . . . . . . . . 14 ⊢ ∪ {(𝑡𝐵((2nd ‘𝑥) + 1))} = ∪ 𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔
144	142, 143	eqtr3i 2756	. . . . . . . . . . . . 13 ⊢ (𝑡𝐵((2nd ‘𝑥) + 1)) = ∪ 𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔
145	140, 144	sseqtri 4016	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ⊆ ∪ 𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔
146		vex 3466	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
147	13, 14, 146	heiborlem1 37514	. . . . . . . . . . . 12 ⊢ (({(𝑡𝐵((2nd ‘𝑥) + 1))} ∈ Fin ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ⊆ ∪ 𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔 ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → ∃𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔 ∈ 𝐾)
148	139, 145, 147	mp3an12 1448	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 → ∃𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔 ∈ 𝐾)
149		eleq1 2814	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑡𝐵((2nd ‘𝑥) + 1)) → (𝑔 ∈ 𝐾 ↔ (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ 𝐾))
150	141, 149	rexsn 4691	. . . . . . . . . . 11 ⊢ (∃𝑔 ∈ {(𝑡𝐵((2nd ‘𝑥) + 1))}𝑔 ∈ 𝐾 ↔ (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ 𝐾)
151	148, 150	sylib 217	. . . . . . . . . 10 ⊢ (((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 → (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ 𝐾)
152		ovex 7459	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥) + 1) ∈ V
153	13, 14, 6, 42, 152	heiborlem2 37515	. . . . . . . . . . 11 ⊢ (𝑡𝐺((2nd ‘𝑥) + 1) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ0 ∧ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ 𝐾))
154	153	biimpri 227	. . . . . . . . . 10 ⊢ ((((2nd ‘𝑥) + 1) ∈ ℕ0 ∧ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ (𝑡𝐵((2nd ‘𝑥) + 1)) ∈ 𝐾) → 𝑡𝐺((2nd ‘𝑥) + 1))
155	137, 138, 151, 154	syl3an 1157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → 𝑡𝐺((2nd ‘𝑥) + 1))
156	155	3expb 1117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → 𝑡𝐺((2nd ‘𝑥) + 1))
157		simprr 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)
158	136, 156, 157	jca32 514	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ (𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → (𝑡 ∈ ∪ 𝐽 ∧ (𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)))
159	158	ex 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ((𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1)) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → (𝑡 ∈ ∪ 𝐽 ∧ (𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))))
160	159	reximdv2 3154	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (∃𝑡 ∈ (𝐹‘((2nd ‘𝑥) + 1))((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 → ∃𝑡 ∈ ∪ 𝐽(𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)))
161	128, 160	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑡 ∈ ∪ 𝐽(𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
162	161	ralrimiva 3136	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ∃𝑡 ∈ ∪ 𝐽(𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
163	13	fvexi 6917	. . . . 5 ⊢ 𝐽 ∈ V
164	163	uniex 7754	. . . 4 ⊢ ∪ 𝐽 ∈ V
165		breq1 5158	. . . . 5 ⊢ (𝑡 = (𝑔‘𝑥) → (𝑡𝐺((2nd ‘𝑥) + 1) ↔ (𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1)))
166		oveq1 7433	. . . . . . 7 ⊢ (𝑡 = (𝑔‘𝑥) → (𝑡𝐵((2nd ‘𝑥) + 1)) = ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1)))
167	166	ineq2d 4213	. . . . . 6 ⊢ (𝑡 = (𝑔‘𝑥) → ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))))
168	167	eleq1d 2811	. . . . 5 ⊢ (𝑡 = (𝑔‘𝑥) → (((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
169	165, 168	anbi12d 630	. . . 4 ⊢ (𝑡 = (𝑔‘𝑥) → ((𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)))
170	164, 169	axcc4dom 10486	. . 3 ⊢ ((𝐺 ≼ ω ∧ ∀𝑥 ∈ 𝐺 ∃𝑡 ∈ ∪ 𝐽(𝑡𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ (𝑡𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → ∃𝑔(𝑔:𝐺⟶∪ 𝐽 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)))
171	58, 162, 170	syl2anc 582	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:𝐺⟶∪ 𝐽 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)))
172		exsimpr 1865	. 2 ⊢ (∃𝑔(𝑔:𝐺⟶∪ 𝐽 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
173	171, 172	syl 17	1 ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))

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  1^st c1st 8003  2^nd c2nd 8004   ≈ cen 8973   ≼ cdom 8974   ≺ csdm 8975  Fincfn 8976  1c1 11161   + caddc 11163  ℝ^*cxr 11299   / cdiv 11923  ℕcn 12266  2c2 12321  ℕ₀cn0 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