Theorem uncmp 22554

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 764	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Top)
2		simpll 764	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝐽 ∈ Top)
3		ssun1 4106	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (𝑆 ∪ 𝑇)
4		sseq2 3947	. . . . . . . . . 10 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → (𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ (𝑆 ∪ 𝑇)))
5	3, 4	mpbiri 257	. . . . . . . . 9 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → 𝑆 ⊆ 𝑋)
6	5	ad2antlr 724	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑆 ⊆ 𝑋)
7		uncmp.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
8	7	cmpsub 22551	. . . . . . . 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 584	. . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑋 = ∪ 𝑐)
11	6, 10	sseqtrd 3961	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑆 ⊆ ∪ 𝑐)
12		unieq 4850	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑐 → ∪ 𝑚 = ∪ 𝑐)
13	12	sseq2d 3953	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑐 → (𝑆 ⊆ ∪ 𝑚 ↔ 𝑆 ⊆ ∪ 𝑐))
14		pweq 4549	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐)
15	14	ineq1d 4145	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑐 → (𝒫 𝑚 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
16	15	rexeqdv 3349	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑐 → (∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛 ↔ ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛))
17	13, 16	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑚 = 𝑐 → ((𝑆 ⊆ ∪ 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 ⊆ ∪ 𝑛) ↔ (𝑆 ⊆ ∪ 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛)))
18	17	rspcv 3557	. . . . . . . . 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 725	. . . . . . . 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 44	. . . . . . 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 239	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑆) ∈ Comp → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑛))
22		ssun2 4107	. . . . . . . . . 10 ⊢ 𝑇 ⊆ (𝑆 ∪ 𝑇)
23		sseq2 3947	. . . . . . . . . 10 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → (𝑇 ⊆ 𝑋 ↔ 𝑇 ⊆ (𝑆 ∪ 𝑇)))
24	22, 23	mpbiri 257	. . . . . . . . 9 ⊢ (𝑋 = (𝑆 ∪ 𝑇) → 𝑇 ⊆ 𝑋)
25	24	ad2antlr 724	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑇 ⊆ 𝑋)
26	7	cmpsub 22551	. . . . . . . 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 584	. . . . . . 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 3961	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → 𝑇 ⊆ ∪ 𝑐)
29		unieq 4850	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑐 → ∪ 𝑟 = ∪ 𝑐)
30	29	sseq2d 3953	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑐 → (𝑇 ⊆ ∪ 𝑟 ↔ 𝑇 ⊆ ∪ 𝑐))
31		pweq 4549	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐)
32	31	ineq1d 4145	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑐 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
33	32	rexeqdv 3349	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑐 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
34	30, 33	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑟 = 𝑐 → ((𝑇 ⊆ ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 ⊆ ∪ 𝑠) ↔ (𝑇 ⊆ ∪ 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠)))
35	34	rspcv 3557	. . . . . . . . 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 725	. . . . . . . 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 44	. . . . . . 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 239	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐)) → ((𝐽 ↾t 𝑇) ∈ Comp → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 ⊆ ∪ 𝑠))
39		reeanv 3294	. . . . . . 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		elinel1 4129	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ 𝒫 𝑐)
41	40	elpwid 4544	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ⊆ 𝑐)
42		elinel1 4129	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ 𝒫 𝑐)
43	42	elpwid 4544	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ⊆ 𝑐)
44	41, 43	anim12i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐))
45	44	ad2antrl 725	. . . . . . . . . . . . 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))
46		unss 4118	. . . . . . . . . . . . 13 ⊢ ((𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐) ↔ (𝑛 ∪ 𝑠) ⊆ 𝑐)
47	45, 46	sylib 217	. . . . . . . . . . . 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)
48		elinel2 4130	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ Fin)
49		elinel2 4130	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ Fin)
50		unfi 8955	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ Fin ∧ 𝑠 ∈ Fin) → (𝑛 ∪ 𝑠) ∈ Fin)
51	48, 49, 50	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛 ∪ 𝑠) ∈ Fin)
52	51	ad2antrl 725	. . . . . . . . . . . 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)
53	47, 52	jca 512	. . . . . . . . . . 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))
54		elin 3903	. . . . . . . . . . . 12 ⊢ ((𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin) ↔ ((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin))
55		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
56	55	elpw2 5269	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ↔ (𝑛 ∪ 𝑠) ⊆ 𝑐)
57	56	anbi1i 624	. . . . . . . . . . . 12 ⊢ (((𝑛 ∪ 𝑠) ∈ 𝒫 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin) ↔ ((𝑛 ∪ 𝑠) ⊆ 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin))
58	54, 57	bitr2i 275	. . . . . . . . . . 11 ⊢ (((𝑛 ∪ 𝑠) ⊆ 𝑐 ∧ (𝑛 ∪ 𝑠) ∈ Fin) ↔ (𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin))
59	53, 58	sylib 217	. . . . . . . . . 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))
60		simpllr 773	. . . . . . . . . . . . 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))
61		ssun3 4108	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ ∪ 𝑛 → 𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
62		ssun4 4109	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ⊆ ∪ 𝑠 → 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
63	61, 62	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠) → (𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠) ∧ 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠)))
64	63	ad2antll 726	. . . . . . . . . . . . . 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)))
65		unss 4118	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (∪ 𝑛 ∪ ∪ 𝑠) ∧ 𝑇 ⊆ (∪ 𝑛 ∪ ∪ 𝑠)) ↔ (𝑆 ∪ 𝑇) ⊆ (∪ 𝑛 ∪ ∪ 𝑠))
66	64, 65	sylib 217	. . . . . . . . . . . . 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))
67	60, 66	eqsstrd 3959	. . . . . . . . . . . 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))
68		uniun 4864	. . . . . . . . . . . 12 ⊢ ∪ (𝑛 ∪ 𝑠) = (∪ 𝑛 ∪ ∪ 𝑠)
69	67, 68	sseqtrrdi 3972	. . . . . . . . . . 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))
70		elpwi 4542	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽)
71	70	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐) → 𝑐 ⊆ 𝐽)
72	71	ad2antlr 724	. . . . . . . . . . . . 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)
73	47, 72	sstrd 3931	. . . . . . . . . . . 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)
74		uniss 4847	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∪ 𝑠) ⊆ 𝐽 → ∪ (𝑛 ∪ 𝑠) ⊆ ∪ 𝐽)
75	74, 7	sseqtrrdi 3972	. . . . . . . . . . . 12 ⊢ ((𝑛 ∪ 𝑠) ⊆ 𝐽 → ∪ (𝑛 ∪ 𝑠) ⊆ 𝑋)
76	73, 75	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)
77	69, 76	eqssd 3938	. . . . . . . . . 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))
78		unieq 4850	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑛 ∪ 𝑠) → ∪ 𝑑 = ∪ (𝑛 ∪ 𝑠))
79	78	rspceeqv 3575	. . . . . . . . . 10 ⊢ (((𝑛 ∪ 𝑠) ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ (𝑛 ∪ 𝑠)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
80	59, 77, 79	syl2anc 584	. . . . . . . . 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)
81	80	exp32 421	. . . . . . . 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)))
82	81	rexlimdvv 3222	. . . . . . 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))
83	39, 82	syl5bir 242	. . . . . 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))
84	21, 38, 83	syl2and 608	. . . . 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))
85	84	impancom 452	. . . 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))
86	85	expd 416	. . 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)))
87	86	ralrimiv 3102	. 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))
88	7	iscmp 22539	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
89	1, 87, 88	sylanbrc 583	1 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  (class class class)co 7275  Fincfn 8733   ↾_t crest 17131  Topctop 22042  Compccmp 22537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538

This theorem is referenced by:  fiuncmp  22555