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Theorem cmpcld 22010

Description: A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)

**Assertion**
Ref	Expression
cmpcld	⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝑆) ∈ Comp)

Proof of Theorem cmpcld Dummy variables 𝑡 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velpw 4544	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐽 ↔ 𝑠 ⊆ 𝐽)
2		simp1l 1193	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝐽 ∈ Comp)
3		simp2 1133	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑠 ⊆ 𝐽)
4		eqid 2821	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 21639	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
6	5	adantl 484	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
7	6	3ad2ant1 1129	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
8	7	snssd 4742	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → {(∪ 𝐽 ∖ 𝑆)} ⊆ 𝐽)
9	3, 8	unssd 4162	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝐽)
10		simp3 1134	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝑠)
11		uniss 4846	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝐽 → ∪ 𝑠 ⊆ ∪ 𝐽)
12	11	3ad2ant2 1130	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝑠 ⊆ ∪ 𝐽)
13	10, 12	sstrd 3977	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝐽)
14		undif 4430	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝐽 ↔ (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
15	13, 14	sylib 220	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
16		unss1 4155	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝑠 → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
17	16	3ad2ant3 1131	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
18	15, 17	eqsstrrd 4006	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
19		difss 4108	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽
20		unss 4160	. . . . . . . . . . 11 ⊢ ((∪ 𝑠 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽) ↔ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
21	12, 19, 20	sylanblc 591	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
22	18, 21	eqssd 3984	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
23		uniexg 7466	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Comp → ∪ 𝐽 ∈ V)
24	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽) → ∪ 𝐽 ∈ V)
25	24	3adant3 1128	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ∈ V)
26		difexg 5231	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∈ V → (∪ 𝐽 ∖ 𝑆) ∈ V)
27		unisng 4857	. . . . . . . . . . 11 ⊢ ((∪ 𝐽 ∖ 𝑆) ∈ V → ∪ {(∪ 𝐽 ∖ 𝑆)} = (∪ 𝐽 ∖ 𝑆))
28	25, 26, 27	3syl 18	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ {(∪ 𝐽 ∖ 𝑆)} = (∪ 𝐽 ∖ 𝑆))
29	28	uneq2d 4139	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
30	22, 29	eqtr4d 2859	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}))
31		uniun 4861	. . . . . . . 8 ⊢ ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)})
32	30, 31	syl6eqr 2874	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
33	4	cmpcov 21997	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)})) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢)
34	2, 9, 32, 33	syl3anc 1367	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢)
35		elfpw 8826	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin))
36		simp2l 1195	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
37		uncom 4129	. . . . . . . . . . . 12 ⊢ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠)
38	36, 37	sseqtrdi 4017	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠))
39		ssundif 4433	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
40	38, 39	sylib 220	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
41		diffi 8750	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
42	41	ad2antll 727	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
43	42	3adant3 1128	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
44		elfpw 8826	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin))
45	40, 43, 44	sylanbrc 585	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin))
46	10	3ad2ant1 1129	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑠)
47	12	3ad2ant1 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝐽)
48		simp3 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝐽 = ∪ 𝑢)
49	47, 48	sseqtrd 4007	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝑢)
50	46, 49	sstrd 3977	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑢)
51	50	sselda 3967	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ ∪ 𝑢)
52		eluni 4841	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑢 ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢))
53	51, 52	sylib 220	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢))
54		simpl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → 𝑣 ∈ 𝑤)
55	54	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → 𝑣 ∈ 𝑤))
56		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → 𝑤 ∈ 𝑢)
57	56	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → 𝑤 ∈ 𝑢))
58		elndif 4105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑆 → ¬ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))
59	58	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) ∧ 𝑣 ∈ 𝑤) → ¬ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))
60		eleq2 2901	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
61	60	biimpd 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 → 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
62	61	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 → 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))))
63	62	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → (𝑣 ∈ 𝑤 → (𝑤 = (∪ 𝐽 ∖ 𝑆) → 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))))
64	63	imp 409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) ∧ 𝑣 ∈ 𝑤) → (𝑤 = (∪ 𝐽 ∖ 𝑆) → 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
65	59, 64	mtod 200	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) ∧ 𝑣 ∈ 𝑤) → ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆))
66	65	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → (𝑣 ∈ 𝑤 → ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆)))
67	66	adantrd 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆)))
68		velsn 4583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ 𝑤 = (∪ 𝐽 ∖ 𝑆))
69	68	notbii 322	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆))
70	67, 69	syl6ibr 254	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)}))
71	57, 70	jcad 515	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → (𝑤 ∈ 𝑢 ∧ ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)})))
72		eldif 3946	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ (𝑤 ∈ 𝑢 ∧ ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)}))
73	71, 72	syl6ibr 254	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
74	55, 73	jcad 515	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → (𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))))
75	74	eximdv 1918	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢) → ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))))
76	53, 75	mpd 15	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
77	76	ex 415	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑣 ∈ 𝑆 → ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))))
78		eluni 4841	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
79	77, 78	syl6ibr 254	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑣 ∈ 𝑆 → 𝑣 ∈ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
80	79	ssrdv 3973	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
81		unieq 4849	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → ∪ 𝑡 = ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
82	81	sseq2d 3999	. . . . . . . . . 10 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → (𝑆 ⊆ ∪ 𝑡 ↔ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
83	82	rspcev 3623	. . . . . . . . 9 ⊢ (((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
84	45, 80, 83	syl2anc 586	. . . . . . . 8 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
85	35, 84	syl3an2b 1400	. . . . . . 7 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ 𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
86	85	rexlimdv3a 3286	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡))
87	34, 86	mpd 15	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
88	87	3exp 1115	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ⊆ 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
89	1, 88	syl5bi 244	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
90	89	ralrimiv 3181	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡))
91		cmptop 22003	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
92	4	cldss 21637	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
93	4	cmpsub 22008	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝐽 ↾_t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
94	91, 92, 93	syl2an 597	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝐽 ↾_t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
95	90, 94	mpbird 259	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝑆) ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ↾_t crest 16694 Topctop 21501 Clsdccld 21624 Compccmp 21994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cld 21627 df-cmp 21995

This theorem is referenced by: hausllycmp 22102 cldllycmp 22103 txkgen 22260 cmphaushmeo 22408 cnheiborlem 23558 cmpcmet 23922 stoweidlem28 42333 stoweidlem50 42355 stoweidlem57 42362

W3C validator