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Theorem cmpcld 23358

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 4561	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐽 ↔ 𝑠 ⊆ 𝐽)
2		simp1l 1199	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝐽 ∈ Comp)
3		simp2 1138	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑠 ⊆ 𝐽)
4		eqid 2737	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 22987	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
6	5	adantl 481	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
7	6	3ad2ant1 1134	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
8	7	snssd 4767	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → {(∪ 𝐽 ∖ 𝑆)} ⊆ 𝐽)
9	3, 8	unssd 4146	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝐽)
10		simp3 1139	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝑠)
11		uniss 4873	. . . . . . . . . . . . . 14 ⊢ (𝑠 ⊆ 𝐽 → ∪ 𝑠 ⊆ ∪ 𝐽)
12	11	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝑠 ⊆ ∪ 𝐽)
13	10, 12	sstrd 3946	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → 𝑆 ⊆ ∪ 𝐽)
14		undif 4436	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝐽 ↔ (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
15	13, 14	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) = ∪ 𝐽)
16		unss1 4139	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ∪ 𝑠 → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
17	16	3ad2ant3 1136	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (𝑆 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
18	15, 17	eqsstrrd 3971	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ⊆ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
19		difss 4090	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽
20		unss 4144	. . . . . . . . . . 11 ⊢ ((∪ 𝑠 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ⊆ ∪ 𝐽) ↔ (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
21	12, 19, 20	sylanblc 590	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)) ⊆ ∪ 𝐽)
22	18, 21	eqssd 3953	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
23		uniexg 7695	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Comp → ∪ 𝐽 ∈ V)
24	23	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽) → ∪ 𝐽 ∈ V)
25	24	3adant3 1133	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 ∈ V)
26		difexg 5276	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∈ V → (∪ 𝐽 ∖ 𝑆) ∈ V)
27		unisng 4883	. . . . . . . . . . 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 4122	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ (∪ 𝐽 ∖ 𝑆)))
30	22, 29	eqtr4d 2775	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)}))
31		uniun 4888	. . . . . . . 8 ⊢ ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = (∪ 𝑠 ∪ ∪ {(∪ 𝐽 ∖ 𝑆)})
32	30, 31	eqtr4di 2790	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∪ 𝐽 = ∪ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
33	4	cmpcov 23345	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin)∪ 𝐽 = ∪ 𝑢)
35		elfpw 9266	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin))
36		simp2l 1201	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}))
37		uncom 4112	. . . . . . . . . . . 12 ⊢ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) = ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠)
38	36, 37	sseqtrdi 3976	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠))
39		ssundif 4442	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ ({(∪ 𝐽 ∖ 𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
40	38, 39	sylib 218	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠)
41		diffi 9111	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
42	41	ad2antll 730	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
43	42	3adant3 1133	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin)
44		elfpw 9266	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ Fin))
45	40, 43, 44	sylanbrc 584	. . . . . . . . 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 1134	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑠)
47	12	3ad2ant1 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝐽)
48		simp3 1139	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝐽 = ∪ 𝑢)
49	47, 48	sseqtrd 3972	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → ∪ 𝑠 ⊆ ∪ 𝑢)
50	46, 49	sstrd 3946	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ 𝑢)
51	50	sselda 3935	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ ∪ 𝑢)
52		eluni 4868	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑢 ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑢))
53	51, 52	sylib 218	. . . . . . . . . . . . 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 4087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑆 → ¬ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆))
59	58	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (∪ 𝐽 ∖ 𝑆) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ (∪ 𝐽 ∖ 𝑆)))
61	60	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . 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 198	. . . . . . . . . . . . . . . . . . . 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 4598	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ 𝑤 = (∪ 𝐽 ∖ 𝑆))
69	68	notbii 320	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)} ↔ ¬ 𝑤 = (∪ 𝐽 ∖ 𝑆))
70	67, 69	imbitrrdi 252	. . . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ (𝑤 ∈ 𝑢 ∧ ¬ 𝑤 ∈ {(∪ 𝐽 ∖ 𝑆)}))
73	71, 72	imbitrrdi 252	. . . . . . . . . . . . . . 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 1919	. . . . . . . . . . . . 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 4868	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
79	77, 78	imbitrrdi 252	. . . . . . . . . 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 3941	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠 ⊆ 𝐽 ∧ 𝑆 ⊆ ∪ 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {(∪ 𝐽 ∖ 𝑆)}) ∧ 𝑢 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑢) → 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
81		unieq 4876	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → ∪ 𝑡 = ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}))
82	81	sseq2d 3968	. . . . . . . . . 10 ⊢ (𝑡 = (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) → (𝑆 ⊆ ∪ 𝑡 ↔ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})))
83	82	rspcev 3578	. . . . . . . . 9 ⊢ (((𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 ⊆ ∪ (𝑢 ∖ {(∪ 𝐽 ∖ 𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)
84	45, 80, 83	syl2anc 585	. . . . . . . 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 1407	. . . . . . 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 3143	. . . . . 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 1120	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ⊆ 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
89	1, 88	biimtrid 242	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡)))
90	89	ralrimiv 3129	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 ⊆ ∪ 𝑡))
91		cmptop 23351	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
92	4	cldss 22985	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
93	4	cmpsub 23356	. . 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 257	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝑆) ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 𝒫 cpw 4556 {csn 4582 ∪ cuni 4865 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ↾_t crest 17352 Topctop 22849 Clsdccld 22972 Compccmp 23342

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-1o 8407 df-en 8896 df-dom 8897 df-fin 8899 df-fi 9326 df-rest 17354 df-topgen 17375 df-top 22850 df-topon 22867 df-bases 22902 df-cld 22975 df-cmp 23343

This theorem is referenced by: hausllycmp 23450 cldllycmp 23451 txkgen 23608 cmphaushmeo 23756 cnheiborlem 24921 cmpcmet 25287 stoweidlem28 46383 stoweidlem50 46405 stoweidlem57 46412

W3C validator