Theorem heiborlem10 38324

Description: Lemma for heibor 38325. The last remaining piece of the proof is to find an element 𝐶 such that 𝐶𝐺0, i.e. 𝐶 is an element of (𝐹‘0) that has no finite subcover, which is true by heiborlem1 38315, since (𝐹‘0) is a finite cover of 𝑋, which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of 𝑈 that covers 𝑋, i.e. 𝑋 is compact. (Contributed by Jeff Madsen, 22-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
heiborlem10	⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣)

Distinct variable groups:   𝑦,𝑛,𝑢,𝐹   𝑚,𝑛,𝑢,𝑣,𝑦,𝑧,𝐷   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑦,𝑧   𝑛,𝐾,𝑦,𝑧   𝜑,𝑣

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)

Proof of Theorem heiborlem10

Dummy variables 𝑡 𝑥 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.7	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
2		0nn0 12498	. . . . . . . 8 ⊢ 0 ∈ ℕ0
3		inss2 4191	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ Fin
4		ffvelcdm 7064	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
5	3, 4	sselid 3936	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
6	1, 2, 5	sylancl 595	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ Fin)
7		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
8		fveq2 6869	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝐹‘𝑛) = (𝐹‘0))
9		oveq2 7406	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
10	8, 9	iuneq12d 4981	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
11	10	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
12	11	rspccva 3582	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
13	7, 2, 12	sylancl 595	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14		eqimss 3996	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) → 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
16		heibor.1	. . . . . . . . . 10 ⊢ 𝐽 = (MetOpen‘𝐷)
17		heibor.3	. . . . . . . . . 10 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
18		ovex 7431	. . . . . . . . . 10 ⊢ (𝑦𝐵0) ∈ V
19	16, 17, 18	heiborlem1 38315	. . . . . . . . 9 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20		oveq1 7405	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
21	20	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
22	21	cbvrexvw 3243	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
23	19, 22	sylib 220	. . . . . . . 8 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24	23	3expia 1135	. . . . . . 7 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
25	6, 15, 24	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
26	25	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
27		heibor.4	. . . . . . . . . 10 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
28		vex 3460	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		c0ex 11175	. . . . . . . . . 10 ⊢ 0 ∈ V
30	16, 17, 27, 28, 29	heiborlem2 38316	. . . . . . . . 9 ⊢ (𝑥𝐺0 ↔ (0 ∈ ℕ0 ∧ 𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾))
31		heibor.5	. . . . . . . . . . . 12 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
32		heibor.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
33	16, 17, 27, 31, 32, 1, 7	heiborlem3 38317	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
34	33	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
35	32	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
36	1	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
37	7	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
38		simprr 782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
39		fveq2 6869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
40		fveq2 6869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (2nd ‘𝑥) = (2nd ‘𝑡))
41	40	oveq1d 7413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((2nd ‘𝑥) + 1) = ((2nd ‘𝑡) + 1))
42	39, 41	breq12d 5115	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ (𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1)))
43		fveq2 6869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (𝐵‘𝑥) = (𝐵‘𝑡))
44	39, 41	oveq12d 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1)) = ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1)))
45	43, 44	ineq12d 4175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))))
46	45	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))) ∈ 𝐾))
47	42, 46	anbi12d 641	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1) ∧ ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))) ∈ 𝐾)))
48	47	cbvralvw 3242	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ ∀𝑡 ∈ 𝐺 ((𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1) ∧ ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))) ∈ 𝐾))
49	38, 48	sylib 220	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → ∀𝑡 ∈ 𝐺 ((𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1) ∧ ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))) ∈ 𝐾))
50		simprl 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51		eqeq1 2768	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52		oveq1 7405	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
53	51, 52	ifbieq2d 4509	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
54	53	cbvmptv 5206	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55		seqeq3 14021	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1))) → seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1)))) = seq0(𝑔, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))))
56	54, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1)))) = seq0(𝑔, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1))))
57		eqid 2764	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ ⟨(seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))))‘𝑛), (3 / (2↑𝑛))⟩) = (𝑛 ∈ ℕ ↦ ⟨(seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))))‘𝑛), (3 / (2↑𝑛))⟩)
58		simplrl 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → 𝑈 ⊆ 𝐽)
59		cmetmet 25350	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60		metxmet 24396	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61	16	mopnuni 24503	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
62	32, 59, 60, 61	4syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
63	62	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑋 = ∪ 𝐽)
64		simprr 782	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝐽 = ∪ 𝑈)
65	63, 64	eqtr2d 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝑈 = 𝑋)
66	65	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → ∪ 𝑈 = 𝑋)
67	16, 17, 27, 31, 35, 36, 37, 49, 50, 56, 57, 58, 66	heiborlem9 38323	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))) → ¬ 𝑋 ∈ 𝐾)
68	67	expr 460	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑥𝐺0) → (∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋 ∈ 𝐾))
69	68	exlimdv 1955	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑥𝐺0) → (∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋 ∈ 𝐾))
70	34, 69	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑥𝐺0) → ¬ 𝑋 ∈ 𝐾)
71	30, 70	sylan2br 604	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (0 ∈ ℕ0 ∧ 𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋 ∈ 𝐾)
72	71	3exp2 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (0 ∈ ℕ0 → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))))
73	2, 72	mpi 20	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾)))
74	73	rexlimdv 3163	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))
75	26, 74	syld 47	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))
76	75	pm2.01d 191	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ¬ 𝑋 ∈ 𝐾)
77		elfvdm 6903	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78		sseq1 3963	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝑋 ⊆ ∪ 𝑣))
79	78	rexbidv 3188	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
80	79	notbid 320	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
81	80, 17	elab2g 3641	. . . . . 6 ⊢ (𝑋 ∈ dom CMet → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
82	32, 77, 81	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
83	82	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
84	83	con2bid 356	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ¬ 𝑋 ∈ 𝐾))
85	76, 84	mpbird 259	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣)
86	62	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = ∪ 𝐽)
87	86	sseq1d 3969	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 ⊆ ∪ 𝑣))
88		inss1 4190	. . . . . . . . 9 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
89	88	sseli 3934	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
90	89	elpwid 4566	. . . . . . 7 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ⊆ 𝑈)
91		simprl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑈 ⊆ 𝐽)
92		sstr 3946	. . . . . . . 8 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → 𝑣 ⊆ 𝐽)
93	92	unissd 4877	. . . . . . 7 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → ∪ 𝑣 ⊆ ∪ 𝐽)
94	90, 91, 93	syl2anr 606	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ∪ 𝑣 ⊆ ∪ 𝐽)
95	94	biantrud 539	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽)))
96		eqss 3953	. . . . 5 ⊢ (∪ 𝐽 = ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽))
97	95, 96	bitr4di 291	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
98	87, 97	bitrd 281	. . 3 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
99	98	rexbidva 3186	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
100	85, 99	mpbid 234	1 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088   ∩ cin 3905   ⊆ wss 3906  ifcif 4482  𝒫 cpw 4557  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5102  {copab 5164   ↦ cmpt 5183  dom cdm 5649  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  2^nd c2nd 7971  Fincfn 8929  0cc0 11075  1c1 11076   + caddc 11078   − cmin 11416   / cdiv 11846  ℕcn 12212  2c2 12274  3c3 12275  ℕ₀cn0 12483  seqcseq 14016  ↑cexp 14076  ∞Metcxmet 21411  Metcmet 21412  ballcbl 21413  MetOpencmopn 21416  CMetccmet 25318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cc 10394  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-pm 8813  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9899  df-acn 9902  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-q 12952  df-rp 12996  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ico 13357  df-icc 13358  df-fl 13804  df-seq 14017  df-exp 14077  df-rest 17453  df-topgen 17474  df-psmet 21418  df-xmet 21419  df-met 21420  df-bl 21421  df-mopn 21422  df-fbas 21423  df-fg 21424  df-top 22956  df-topon 22973  df-bases 23008  df-cld 23081  df-ntr 23082  df-cls 23083  df-nei 23160  df-lm 23291  df-haus 23377  df-fil 23908  df-fm 24000  df-flim 24001  df-flf 24002  df-cfil 25319  df-cau 25320  df-cmet 25321

This theorem is referenced by:  heibor  38325