Theorem heiborlem10 37827

Description: Lemma for heibor 37828. 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 37818, 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 12541	. . . . . . . 8 ⊢ 0 ∈ ℕ0
3		inss2 4238	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ Fin
4		ffvelcdm 7101	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
5	3, 4	sselid 3981	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
6	1, 2, 5	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ Fin)
7		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
8		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝐹‘𝑛) = (𝐹‘0))
9		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
10	8, 9	iuneq12d 5021	. . . . . . . . . . 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 3621	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
13	7, 2, 12	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14		eqimss 4042	. . . . . . . 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 7464	. . . . . . . . . 10 ⊢ (𝑦𝐵0) ∈ V
19	16, 17, 18	heiborlem1 37818	. . . . . . . . 9 ⊢ (((𝐹‘0) ∈ Fin ∧ 𝑋 ⊆ ∪ 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋 ∈ 𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
21	20	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
22	21	cbvrexvw 3238	. . . . . . . . 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 584	. . . . . 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 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		c0ex 11255	. . . . . . . . . 10 ⊢ 0 ∈ V
30	16, 17, 27, 28, 29	heiborlem2 37819	. . . . . . . . 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 37820	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
34	33	ad2antrr 726	. . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
40		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (2nd ‘𝑥) = (2nd ‘𝑡))
41	40	oveq1d 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((2nd ‘𝑥) + 1) = ((2nd ‘𝑡) + 1))
42	39, 41	breq12d 5156	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ (𝑔‘𝑡)𝐺((2nd ‘𝑡) + 1)))
43		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (𝐵‘𝑥) = (𝐵‘𝑡))
44	39, 41	oveq12d 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1)) = ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1)))
45	43, 44	ineq12d 4221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑡 → ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) = ((𝐵‘𝑡) ∩ ((𝑔‘𝑡)𝐵((2nd ‘𝑡) + 1))))
46	45	eleq1d 2826	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . 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 3237	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
53	51, 52	ifbieq2d 4552	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
54	53	cbvmptv 5255	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55		seqeq3 14047	. . . . . . . . . . . . . 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 25320	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60		metxmet 24344	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61	16	mopnuni 24451	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 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 37826	. . . . . . . . . . . 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 1933	. . . . . . . . . 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 595	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ (0 ∈ ℕ0 ∧ 𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋 ∈ 𝐾)
72	71	3exp2 1355	. . . . . . 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 3153	. . . . 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 6943	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78		sseq1 4009	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝑋 ⊆ ∪ 𝑣))
79	78	rexbidv 3179	. . . . . . . 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 3680	. . . . . 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 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = ∪ 𝐽)
87	86	sseq1d 4015	. . . 4 ⊢ (((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 ⊆ ∪ 𝑣 ↔ ∪ 𝐽 ⊆ ∪ 𝑣))
88		inss1 4237	. . . . . . . . 9 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
89	88	sseli 3979	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
90	89	elpwid 4609	. . . . . . 7 ⊢ (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ⊆ 𝑈)
91		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → 𝑈 ⊆ 𝐽)
92		sstr 3992	. . . . . . . 8 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → 𝑣 ⊆ 𝐽)
93	92	unissd 4917	. . . . . . 7 ⊢ ((𝑣 ⊆ 𝑈 ∧ 𝑈 ⊆ 𝐽) → ∪ 𝑣 ⊆ ∪ 𝐽)
94	90, 91, 93	syl2anr 597	. . . . . 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 3999	. . . . 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 3177	. 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 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  ifcif 4525  𝒫 cpw 4600  ⟨cop 4632  ∪ cuni 4907  ∪ ciun 4991   class class class wbr 5143  {copab 5205   ↦ cmpt 5225  dom cdm 5685  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  2^nd c2nd 8013  Fincfn 8985  0cc0 11155  1c1 11156   + caddc 11158   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  3c3 12322  ℕ₀cn0 12526  seqcseq 14042  ↑cexp 14102  ∞Metcxmet 21349  Metcmet 21350  ballcbl 21351  MetOpencmopn 21354  CMetccmet 25288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-icc 13394  df-fl 13832  df-seq 14043  df-exp 14103  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lm 23237  df-haus 23323  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-cfil 25289  df-cau 25290  df-cmet 25291

This theorem is referenced by:  heibor  37828