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Theorem cmpcld 23596

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 4572	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐽 ↔ 𝑠 ⊆ 𝐽)
2		simp1l 1216	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝐽 ∈ Comp)
3		simp2 1155	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑠 ⊆ 𝐽)
4		eqid 2766	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 23225	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
6	5	adantl 487	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
7	6	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
8	7	snssd 4757	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → {(∪ 𝐽 ∖ 𝑆)} ⊆ 𝐽)
9	3, 8	unssd 4148	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝐽)
10		simp3 1156	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝑠)
11		uniss 4885	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝐽 → ∪ 𝑠 ⊆ ∪ 𝐽)
12	11	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝑠 ⊆ ∪ 𝐽)
13	10, 12	sstrd 3950	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝐽)
14		undif 4448	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝐽 ↔ (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
15	13, 14	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
16		unss1 4141	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝑠 → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
17	16	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
18	15, 17	eqsstrrd 3975	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
19		difss 4093	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽
20		unss 4146	. . . . . . . . . . 11 ⊢ ((∪ 𝑠 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽) ↔ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
21	12, 19, 20	sylanblc 601	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
22	18, 21	eqssd 3957	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
23		uniexg 7751	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Comp → ∪ 𝐽 ∈ V)
24	23	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽) → ∪ 𝐽 ∈ V)
25	24	3adant3 1150	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ∈ V)
26		difexg 5305	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∈ V → (∪ 𝐽 ∖ 𝑆) ∈ V)
27		unisng 4895	. . . . . . . . . . 11 ⊢ ((∪ 𝐽 ∖ 𝑆) ∈ V → ∪ {(∪ 𝐽 ∖ 𝑆)} = (∪ 𝐽 ∖ 𝑆))
28	25, 26, 27	3syl 19	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ {(∪ 𝐽 ∖ 𝑆)} = (∪ 𝐽 ∖ 𝑆))
29	28	uneq2d 4125	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
30	22, 29	eqtr4d 2804	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}))
31		uniun 4900	. . . . . . . 8 ⊢ ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)})
32	30, 31	eqtr4di 2819	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
33	4	cmpcov 23583	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢)
35		elfpw 9321	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin))
36		simp2l 1218	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
37		uncom 4115	. . . . . . . . . . . 12 ⊢ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠)
38	36, 37	sseqtrdi 3980	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠))
39		ssundif 4453	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
40	38, 39	sylib 221	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
41		diffi 9169	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
42	41	ad2antll 742	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
43	42	3adant3 1150	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
44		elfpw 9321	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin))
45	40, 43, 44	sylanbrc 595	. . . . . . . . 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 1151	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑠)
47	12	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝐽)
48		simp3 1156	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝐽 = ∪ 𝑢)
49	47, 48	sseqtrd 3976	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝑢)
50	46, 49	sstrd 3950	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑢)
51	50	sselda 3940	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ ∪ 𝑢)
52		eluni 4880	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑢 ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢))
53	51, 52	sylib 221	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑆 → ¬ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))
59	58	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
61	60	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . 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 201	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . 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 4610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ 𝑤 = (∪ 𝐽 ∖ 𝑆))
69	68	notbii 323	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆))
70	67, 69	imbitrrdi 255	. . . . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ (𝑤 ∈ 𝑢 ∧ ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)}))
73	71, 72	imbitrrdi 255	. . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . 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 1950	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 4880	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
79	77, 78	imbitrrdi 255	. . . . . . . . . 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 3946	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
81		unieq 4888	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → ∪ 𝑡 = ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
82	81	sseq2d 3972	. . . . . . . . . 10 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → (𝑆 ⊆ ∪ 𝑡 ↔ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
83	82	rspcev 3584	. . . . . . . . 9 ⊢ (((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
84	45, 80, 83	syl2anc 596	. . . . . . . 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 1431	. . . . . . 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 3173	. . . . . 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 16	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
88	87	3exp 1137	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ⊆ 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
89	1, 88	biimtrid 245	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
90	89	ralrimiv 3159	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡))
91		cmptop 23589	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
92	4	cldss 23223	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
93	4	cmpsub 23594	. . 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 608	. 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 260	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝑆) ∈ Comp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ∀wral 3082 ∃wrex 3092 Vcvv 3458 ∖ cdif 3905 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 𝒫 cpw 4567 {csn 4594 ∪ cuni 4877 ‘cfv 6543 (class class class)co 7423 Fincfn 8952 ↾_t crest 17498 Topctop 23087 Clsdccld 23210 Compccmp 23580

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-1o 8462 df-en 8953 df-dom 8954 df-fin 8956 df-fi 9381 df-rest 17500 df-topgen 17521 df-top 23088 df-topon 23105 df-bases 23140 df-cld 23213 df-cmp 23581

This theorem is used by: hausllycmp 23688 cldllycmp 23689 txkgen 23846 cmphaushmeo 23994 cnheiborlem 25150 cmpcmet 25515 stoweidlem28 46783 stoweidlem50 46805 stoweidlem57 46812

W3C validator