Theorem heiborlem10 38415

Description: Lemma for heibor 38416. 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 38406, 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 12518	. . . . . . . 8 ⊢ 0 ∈ ℕ0
3		inss2 4189	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ Fin
4		ffvelcdm 7076	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
5	3, 4	sselid 3934	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
6	1, 2, 5	sylancl 597	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ Fin)
7		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
8		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝐹‘𝑛) = (𝐹‘0))
9		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
10	8, 9	iuneq12d 4985	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
11	10	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
12	11	rspccva 3579	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
13	7, 2, 12	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14		eqimss 3994	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) → 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
15	13, 14	syl 18	. . . . . . 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 7443	. . . . . . . . . 10 ⊢ (𝑦𝐵0) ∈ V
19	16, 17, 18	heiborlem1 38406	. . . . . . . . 9 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
21	20	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
22	21	cbvrexvw 3242	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
23	19, 22	sylib 221	. . . . . . . 8 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24	23	3expia 1137	. . . . . . 7 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
25	6, 15, 24	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
26	25	adantr 485	. . . . 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 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		c0ex 11199	. . . . . . . . . 10 ⊢ 0 ∈ V
30	16, 17, 27, 28, 29	heiborlem2 38407	. . . . . . . . 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 38408	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
34	33	ad2antrr 738	. . . . . . . . . 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 738	. . . . . . . . . . . . 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 738	. . . . . . . . . . . . 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 738	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
40		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (2nd ‘𝑥) = (2nd ‘𝑡))
41	40	oveq1d 7425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((2nd ‘𝑥) + 1) = ((2nd ‘𝑡) + 1))
42	39, 41	breq12d 5121	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ (𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1)))
43		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (𝐵‘𝑥) = (𝐵‘𝑡))
44	39, 41	oveq12d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1)) = ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1)))
45	43, 44	ineq12d 4173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))))
46	45	eleq1d 2846	. . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . 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 3241	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52		oveq1 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
53	51, 52	ifbieq2d 4513	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
54	53	cbvmptv 5214	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55		seqeq3 14041	. . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 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 25424	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60		metxmet 24470	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61	16	mopnuni 24577	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
62	32, 59, 60, 61	4syl 20	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
63	62	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑋 = ∪ 𝐽)
64		simprr 784	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝐽 = ∪ 𝑈)
65	63, 64	eqtr2d 2797	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∪ 𝑈 = 𝑋)
66	65	adantr 485	. . . . . . . . . . . . 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 38414	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 1961	. . . . . . . . . 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 16	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑥𝐺0) → ¬ 𝑋 ∈ 𝐾)
71	30, 70	sylan2br 606	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (0 ∈ ℕ0 ∧ 𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋 ∈ 𝐾)
72	71	3exp2 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (0 ∈ ℕ0 → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))))
73	2, 72	mpi 21	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾)))
74	73	rexlimdv 3162	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))
75	26, 74	syld 48	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 → ¬ 𝑋 ∈ 𝐾))
76	75	pm2.01d 192	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ¬ 𝑋 ∈ 𝐾)
77		elfvdm 6915	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78		sseq1 3961	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝑋 ⊆ ∪ 𝑣))
79	78	rexbidv 3187	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
80	79	notbid 321	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
81	80, 17	elab2g 3638	. . . . . 6 ⊢ (𝑋 ∈ dom CMet → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
82	32, 77, 81	3syl 19	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
83	82	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (𝑋 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣))
84	83	con2bid 357	. . 3 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ¬ 𝑋 ∈ 𝐾))
85	76, 84	mpbird 260	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣)
86	62	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = ∪ 𝐽)
87	86	sseq1d 3967	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 ⊆ ∪ 𝑣))
88		inss1 4188	. . . . . . . . 9 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
89	88	sseli 3932	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
90	89	elpwid 4570	. . . . . . 7 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ⊆ 𝑈)
91		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑈 ⊆ 𝐽)
92		sstr 3944	. . . . . . . 8 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → 𝑣 ⊆ 𝐽)
93	92	unissd 4881	. . . . . . 7 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → ∪ 𝑣 ⊆ ∪ 𝐽)
94	90, 91, 93	syl2anr 608	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ∪ 𝑣 ⊆ ∪ 𝐽)
95	94	biantrud 540	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽)))
96		eqss 3951	. . . . 5 ⊢ (∪ 𝐽 = ∪ 𝑣 ↔ (∪ 𝐽 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ⊆ ∪ 𝐽))
97	95, 96	bitr4di 292	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (∪ 𝐽 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
98	87, 97	bitrd 282	. . 3 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 = ∪ 𝑣))
99	98	rexbidva 3185	. 2 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
100	85, 99	mpbid 235	1 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568  ∃wex 1807   ∈ wcel 2141  {cab 2739  ∀wral 3077  ∃wrex 3087   ∩ cin 3903   ⊆ wss 3904  ifcif 4486  𝒫 cpw 4561  ⟨cop 4594  ∪ cuni 4871  ∪ ciun 4955   class class class wbr 5108  {copab 5172   ↦ cmpt 5191  dom cdm 5661  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  2^nd c2nd 7984  Fincfn 8942  0cc0 11099  1c1 11100   + caddc 11102   − cmin 11440   / cdiv 11870  ℕcn 12232  2c2 12294  3c3 12295  ℕ₀cn0 12503  seqcseq 14036  ↑cexp 14096  ∞Metcxmet 21486  Metcmet 21487  ballcbl 21488  MetOpencmopn 21491  CMetccmet 25392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cc 10418  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-oi 9471  df-card 9924  df-acn 9927  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ico 13377  df-icc 13378  df-fl 13824  df-seq 14037  df-exp 14097  df-rest 17474  df-topgen 17495  df-psmet 21493  df-xmet 21494  df-met 21495  df-bl 21496  df-mopn 21497  df-fbas 21498  df-fg 21499  df-top 23030  df-topon 23047  df-bases 23082  df-cld 23155  df-ntr 23156  df-cls 23157  df-nei 23234  df-lm 23365  df-haus 23451  df-fil 23982  df-fm 24074  df-flim 24075  df-flf 24076  df-cfil 25393  df-cau 25394  df-cmet 25395

This theorem is referenced by:  heibor  38416