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Theorem cmpcld 22534

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 4543	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐽 ↔ 𝑠 ⊆ 𝐽)
2		simp1l 1195	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝐽 ∈ Comp)
3		simp2 1135	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑠 ⊆ 𝐽)
4		eqid 2739	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 22163	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
6	5	adantl 481	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
7	6	3ad2ant1 1131	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
8	7	snssd 4747	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → {(∪ 𝐽 ∖ 𝑆)} ⊆ 𝐽)
9	3, 8	unssd 4124	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝐽)
10		simp3 1136	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝑠)
11		uniss 4852	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝐽 → ∪ 𝑠 ⊆ ∪ 𝐽)
12	11	3ad2ant2 1132	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝑠 ⊆ ∪ 𝐽)
13	10, 12	sstrd 3935	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝐽)
14		undif 4420	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝐽 ↔ (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
15	13, 14	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
16		unss1 4117	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝑠 → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
17	16	3ad2ant3 1133	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
18	15, 17	eqsstrrd 3964	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
19		difss 4070	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽
20		unss 4122	. . . . . . . . . . 11 ⊢ ((∪ 𝑠 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽) ↔ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
21	12, 19, 20	sylanblc 588	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
22	18, 21	eqssd 3942	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
23		uniexg 7584	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Comp → ∪ 𝐽 ∈ V)
24	23	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽) → ∪ 𝐽 ∈ V)
25	24	3adant3 1130	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ∈ V)
26		difexg 5254	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∈ V → (∪ 𝐽 ∖ 𝑆) ∈ V)
27		unisng 4865	. . . . . . . . . . 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 4101	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
30	22, 29	eqtr4d 2782	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}))
31		uniun 4869	. . . . . . . 8 ⊢ ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)})
32	30, 31	eqtr4di 2797	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
33	4	cmpcov 22521	. . . . . . 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 1369	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢)
35		elfpw 9082	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin))
36		simp2l 1197	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
37		uncom 4091	. . . . . . . . . . . 12 ⊢ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠)
38	36, 37	sseqtrdi 3975	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠))
39		ssundif 4423	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
40	38, 39	sylib 217	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
41		diffi 8927	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
42	41	ad2antll 725	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
43	42	3adant3 1130	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
44		elfpw 9082	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin))
45	40, 43, 44	sylanbrc 582	. . . . . . . . 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 1131	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑠)
47	12	3ad2ant1 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝐽)
48		simp3 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝐽 = ∪ 𝑢)
49	47, 48	sseqtrd 3965	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝑢)
50	46, 49	sstrd 3935	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑢)
51	50	sselda 3925	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ ∪ 𝑢)
52		eluni 4847	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑢 ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢))
53	51, 52	sylib 217	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 4067	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑆 → ¬ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))
59	58	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
61	60	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . 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 197	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . . 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 4582	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ 𝑤 = (∪ 𝐽 ∖ 𝑆))
69	68	notbii 319	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆))
70	67, 69	syl6ibr 251	. . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 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 3901	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ (𝑤 ∈ 𝑢 ∧ ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)}))
73	71, 72	syl6ibr 251	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . 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 1923	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 4847	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
79	77, 78	syl6ibr 251	. . . . . . . . . 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 3931	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
81		unieq 4855	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → ∪ 𝑡 = ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
82	81	sseq2d 3957	. . . . . . . . . 10 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → (𝑆 ⊆ ∪ 𝑡 ↔ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
83	82	rspcev 3560	. . . . . . . . 9 ⊢ (((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
84	45, 80, 83	syl2anc 583	. . . . . . . 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 1402	. . . . . . 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 3216	. . . . . 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 1117	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ⊆ 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
89	1, 88	syl5bi 241	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
90	89	ralrimiv 3108	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡))
91		cmptop 22527	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
92	4	cldss 22161	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
93	4	cmpsub 22532	. . 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 595	. 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 256	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝑆) ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∃wex 1785 ∈ wcel 2109 ∀wral 3065 ∃wrex 3066 Vcvv 3430 ∖ cdif 3888 ∪ cun 3889 ∩ cin 3890 ⊆ wss 3891 𝒫 cpw 4538 {csn 4566 ∪ cuni 4844 ‘cfv 6430 (class class class)co 7268 Fincfn 8707 ↾_t crest 17112 Topctop 22023 Clsdccld 22148 Compccmp 22518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-fin 8711 df-fi 9131 df-rest 17114 df-topgen 17135 df-top 22024 df-topon 22041 df-bases 22077 df-cld 22151 df-cmp 22519

This theorem is referenced by: hausllycmp 22626 cldllycmp 22627 txkgen 22784 cmphaushmeo 22932 cnheiborlem 24098 cmpcmet 24464 stoweidlem28 43523 stoweidlem50 43545 stoweidlem57 43552

W3C validator