Theorem heiborlem10 38023

Description: Lemma for heibor 38024. 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 38014, 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 12420	. . . . . . . 8 ⊢ 0 ∈ ℕ0
3		inss2 4191	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ Fin
4		ffvelcdm 7028	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
5	3, 4	sselid 3932	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
6	1, 2, 5	sylancl 587	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ Fin)
7		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
8		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝐹‘𝑛) = (𝐹‘0))
9		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
10	8, 9	iuneq12d 4977	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
11	10	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
12	11	rspccva 3576	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
13	7, 2, 12	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14		eqimss 3993	. . . . . . . 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 7393	. . . . . . . . . 10 ⊢ (𝑦𝐵0) ∈ V
19	16, 17, 18	heiborlem1 38014	. . . . . . . . 9 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
21	20	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
22	21	cbvrexvw 3216	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
23	19, 22	sylib 218	. . . . . . . 8 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24	23	3expia 1122	. . . . . . 7 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
25	6, 15, 24	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
26	25	adantr 480	. . . . 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 3445	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		c0ex 11130	. . . . . . . . . 10 ⊢ 0 ∈ V
30	16, 17, 27, 28, 29	heiborlem2 38015	. . . . . . . . 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 38016	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
34	33	ad2antrr 727	. . . . . . . . . 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 727	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
40		fveq2 6835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (2nd ‘𝑥) = (2nd ‘𝑡))
41	40	oveq1d 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((2nd ‘𝑥) + 1) = ((2nd ‘𝑡) + 1))
42	39, 41	breq12d 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ (𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1)))
43		fveq2 6835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (𝐵‘𝑥) = (𝐵‘𝑡))
44	39, 41	oveq12d 7378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1)) = ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1)))
45	43, 44	ineq12d 4174	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))))
46	45	eleq1d 2822	. . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52		oveq1 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
53	51, 52	ifbieq2d 4507	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
54	53	cbvmptv 5203	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55		seqeq3 13933	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 25246	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60		metxmet 24282	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61	16	mopnuni 24389	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
62	32, 59, 60, 61	4syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
63	62	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑋 = ∪ 𝐽)
64		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝐽 = ∪ 𝑈)
65	63, 64	eqtr2d 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝑈 = 𝑋)
66	65	adantr 480	. . . . . . . . . . . . 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 38022	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1935	. . . . . . . . . 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 596	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (0 ∈ ℕ0 ∧ 𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋 ∈ 𝐾)
72	71	3exp2 1356	. . . . . . 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 3136	. . . . 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 190	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ¬ 𝑋 ∈ 𝐾)
77		elfvdm 6869	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78		sseq1 3960	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝑋 ⊆ ∪ 𝑣))
79	78	rexbidv 3161	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
80	79	notbid 318	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
81	80, 17	elab2g 3636	. . . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
84	83	con2bid 354	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ¬ 𝑋 ∈ 𝐾))
85	76, 84	mpbird 257	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣)
86	62	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = ∪ 𝐽)
87	86	sseq1d 3966	. . . 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 3930	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
90	89	elpwid 4564	. . . . . . 7 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ⊆ 𝑈)
91		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑈 ⊆ 𝐽)
92		sstr 3943	. . . . . . . 8 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → 𝑣 ⊆ 𝐽)
93	92	unissd 4874	. . . . . . 7 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → ∪ 𝑣 ⊆ ∪ 𝐽)
94	90, 91, 93	syl2anr 598	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ∪ 𝑣 ⊆ ∪ 𝐽)
95	94	biantrud 531	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽)))
96		eqss 3950	. . . . 5 ⊢ (∪ 𝐽 = ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽))
97	95, 96	bitr4di 289	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
98	87, 97	bitrd 279	. . 3 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
99	98	rexbidva 3159	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
100	85, 99	mpbid 232	1 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3061   ∩ cin 3901   ⊆ wss 3902  ifcif 4480  𝒫 cpw 4555  ⟨cop 4587  ∪ cuni 4864  ∪ ciun 4947   class class class wbr 5099  {copab 5161   ↦ cmpt 5180  dom cdm 5625  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  2^nd c2nd 7934  Fincfn 8887  0cc0 11030  1c1 11031   + caddc 11033   − cmin 11368   / cdiv 11798  ℕcn 12149  2c2 12204  3c3 12205  ℕ₀cn0 12405  seqcseq 13928  ↑cexp 13988  ∞Metcxmet 21298  Metcmet 21299  ballcbl 21300  MetOpencmopn 21303  CMetccmet 25214

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cc 10349  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-acn 9858  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-q 12866  df-rp 12910  df-xneg 13030  df-xadd 13031  df-xmul 13032  df-ico 13271  df-icc 13272  df-fl 13716  df-seq 13929  df-exp 13989  df-rest 17346  df-topgen 17367  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-fbas 21310  df-fg 21311  df-top 22842  df-topon 22859  df-bases 22894  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lm 23177  df-haus 23263  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-cfil 25215  df-cau 25216  df-cmet 25217

This theorem is referenced by:  heibor  38024