Theorem uncmp 21016

Description: The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)

Hypothesis

Ref	Expression
uncmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
uncmp	⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)

Proof of Theorem uncmp

Dummy variables 𝑐 𝑑 𝑚 𝑛 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 786	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Top)
2		simpll 786	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝐽 ∈ Top)
3		ssun1 3738	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (𝑆 ∪ 𝑇)
4		sseq2 3590	. . . . . . . . . 10 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → (𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ (𝑆 ∪ 𝑇)))
5	3, 4	mpbiri 247	. . . . . . . . 9 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → 𝑆 ⊆ 𝑋)
6	5	ad2antlr 759	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑆 ⊆ 𝑋)
7		uncmp.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
8	7	cmpsub 21013	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
9	2, 6, 8	syl2anc 691	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
10		simprr 792	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑋 = ∪ 𝑐)
11	6, 10	sseqtrd 3604	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑆 ⊆ ∪ 𝑐)
12		unieq 4380	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑐 → ∪ 𝑚 = ∪ 𝑐)
13	12	sseq2d 3596	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑐 → (𝑆 ⊆ ∪ 𝑚 ↔ 𝑆 ⊆ ∪ 𝑐))
14		pweq 4111	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐)
15	14	ineq1d 3775	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑐 → (𝒫 𝑚 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
16	15	rexeqdv 3122	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑐 → (∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛 ↔ ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛))
17	13, 16	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑚 = 𝑐 → ((𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛) ↔ (𝑆 ⊆ ∪ 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
18	17	rspcv 3278	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝒫 𝐽 → (∀𝑚 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛) → (𝑆 ⊆ ∪ 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
19	18	ad2antrl 760	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (∀𝑚 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛) → (𝑆 ⊆ ∪ 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
20	11, 19	mpid 43	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (∀𝑚 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛) → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛))
21	9, 20	sylbid 229	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑆) ∈ Comp → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛))
22		ssun2 3739	. . . . . . . . . 10 ⊢ 𝑇 ⊆ (𝑆 ∪ 𝑇)
23		sseq2 3590	. . . . . . . . . 10 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → (𝑇 ⊆ 𝑋 ↔ 𝑇 ⊆ (𝑆 ∪ 𝑇)))
24	22, 23	mpbiri 247	. . . . . . . . 9 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → 𝑇 ⊆ 𝑋)
25	24	ad2antlr 759	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑇 ⊆ 𝑋)
26	7	cmpsub 21013	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((𝐽 ↾t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
27	2, 25, 26	syl2anc 691	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
28	25, 10	sseqtrd 3604	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑇 ⊆ ∪ 𝑐)
29		unieq 4380	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑐 → ∪ 𝑟 = ∪ 𝑐)
30	29	sseq2d 3596	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑐 → (𝑇 ⊆ ∪ 𝑟 ↔ 𝑇 ⊆ ∪ 𝑐))
31		pweq 4111	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐)
32	31	ineq1d 3775	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑐 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
33	32	rexeqdv 3122	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑐 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
34	30, 33	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑟 = 𝑐 → ((𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠) ↔ (𝑇 ⊆ ∪ 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
35	34	rspcv 3278	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝒫 𝐽 → (∀𝑟 ∈ 𝒫 𝐽(𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠) → (𝑇 ⊆ ∪ 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
36	35	ad2antrl 760	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (∀𝑟 ∈ 𝒫 𝐽(𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠) → (𝑇 ⊆ ∪ 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
37	28, 36	mpid 43	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (∀𝑟 ∈ 𝒫 𝐽(𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠) → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
38	27, 37	sylbid 229	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑇) ∈ Comp → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
39		reeanv 3086	. . . . . . 7 ⊢ (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)(𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠) ↔ (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
40		elin 3758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑐 ∧ 𝑛 ∈ Fin))
41	40	simplbi 475	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ 𝒫 𝑐)
42	41	elpwid 4118	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ⊆ 𝑐)
43		elin 3758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑠 ∈ 𝒫 𝑐 ∧ 𝑠 ∈ Fin))
44	43	simplbi 475	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ 𝒫 𝑐)
45	44	elpwid 4118	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ⊆ 𝑐)
46	42, 45	anim12i 588	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐))
47	46	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐))
48		unss 3749	. . . . . . . . . . . . 13 ⊢ ((𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐) ↔ (𝑛 ∪ 𝑠) ⊆ 𝑐)
49	47, 48	sylib 207	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑛 ∪ 𝑠) ⊆ 𝑐)
50	40	simprbi 479	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ Fin)
51	43	simprbi 479	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ Fin)
52		unfi 8112	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ Fin ∧ 𝑠 ∈ Fin) → (𝑛 ∪ 𝑠) ∈ Fin)
53	50, 51, 52	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛 ∪ 𝑠) ∈ Fin)
54	53	ad2antrl 760	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑛 ∪ 𝑠) ∈ Fin)
55	49, 54	jca 553	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → ((𝑛 ∪ 𝑠) ⊆ 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin))
56		elin 3758	. . . . . . . . . . . 12 ⊢ ((𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin) ↔ ((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin))
57		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
58	57	elpw2 4755	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ↔ (𝑛 ∪ 𝑠) ⊆ 𝑐)
59	58	anbi1i 727	. . . . . . . . . . . 12 ⊢ (((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin) ↔ ((𝑛 ∪ 𝑠) ⊆ 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin))
60	56, 59	bitr2i 264	. . . . . . . . . . 11 ⊢ (((𝑛 ∪ 𝑠) ⊆ 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin) ↔ (𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin))
61	55, 60	sylib 207	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin))
62		simpllr 795	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → 𝑋 = (𝑆 ∪ 𝑇))
63		ssun3 3740	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ ∪ 𝑛 → 𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
64		ssun4 3741	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ⊆ ∪ 𝑠 → 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
65	63, 64	anim12i 588	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠) → (𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠) ∧ 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠)))
66	65	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠) ∧ 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠)))
67		unss 3749	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠) ∧ 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠)) ↔ (𝑆 ∪ 𝑇) ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
68	66, 67	sylib 207	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑆 ∪ 𝑇) ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
69	62, 68	eqsstrd 3602	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → 𝑋 ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
70		uniun 4392	. . . . . . . . . . . 12 ⊢ ∪ (𝑛 ∪ 𝑠) = (∪ 𝑛 ∪ ∪ 𝑠)
71	69, 70	syl6sseqr 3615	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → 𝑋 ⊆ ∪ (𝑛 ∪ 𝑠))
72		elpwi 4117	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽)
73	72	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐) → 𝑐 ⊆ 𝐽)
74	73	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → 𝑐 ⊆ 𝐽)
75	49, 74	sstrd 3578	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → (𝑛 ∪ 𝑠) ⊆ 𝐽)
76		uniss 4394	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∪ 𝑠) ⊆ 𝐽 → ∪ (𝑛 ∪ 𝑠) ⊆ ∪ 𝐽)
77	76, 7	syl6sseqr 3615	. . . . . . . . . . . 12 ⊢ ((𝑛 ∪ 𝑠) ⊆ 𝐽 → ∪ (𝑛 ∪ 𝑠) ⊆ 𝑋)
78	75, 77	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → ∪ (𝑛 ∪ 𝑠) ⊆ 𝑋)
79	71, 78	eqssd 3585	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → 𝑋 = ∪ (𝑛 ∪ 𝑠))
80		unieq 4380	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑛 ∪ 𝑠) → ∪ 𝑑 = ∪ (𝑛 ∪ 𝑠))
81	80	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑛 ∪ 𝑠) → (𝑋 = ∪ 𝑑 ↔ 𝑋 = ∪ (𝑛 ∪ 𝑠)))
82	81	rspcev 3282	. . . . . . . . . 10 ⊢ (((𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ (𝑛 ∪ 𝑠)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
83	61, 79, 82	syl2anc 691	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠))) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
84	83	exp32 629	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → ((𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
85	84	rexlimdvv 3019	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)(𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
86	39, 85	syl5bir 232	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
87	21, 38, 86	syl2and 499	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → (((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
88	87	impancom 455	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → ((𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
89	88	expd 451	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
90	89	ralrimiv 2948	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
91	7	iscmp 21001	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
92	1, 90, 91	sylanbrc 695	1 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  (class class class)co 6549  Fincfn 7841   ↾_t crest 15904  Topctop 20517  Compccmp 20999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000

This theorem is referenced by:  fiuncmp  21017