Theorem reheibor 16025

Description: Heine-Borel theorem for real numbers. A subset of RR is compact iff it is closed and bounded.

Hypotheses

Ref	Expression
reheibor.1	|- R = ((abs o. - ) |` (RR X. RR))
reheibor.2	|- M = (R |` (Y X. Y))
reheibor.3	|- T = (Open` M)
reheibor.4	|- U = (Open` R)

Assertion

Ref	Expression
reheibor	|- (Y C_ RR -> (T e. Comp <-> (Y e. (Clsd` U) /\ M e. Bnd)))

Proof of Theorem reheibor

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- (RR ^m (1...1)) = (RR ^m (1...1))
2		eqid 1884	. . . 4 |- ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) = ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))
3		eqid 1884	. . . 4 |- (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) = (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))
4		eqid 1884	. . . 4 |- (Open` (RRn` 1)) = (Open` (RRn` 1))
5	1, 2, 3, 4	rrnheibor 16023	. . 3 |- ((1 e. NN /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) C_ (RR ^m (1...1))) -> ((Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp <-> (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) e. (Clsd` (Open` (RRn` 1))) /\ ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Bnd)))
6		1nn 7117	. . 3 |- 1 e. NN
7		imassrn 4278	. . . . 5 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) C_ ran {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}
8		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} = {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}
9		fconstg 4604	. . . . . . . . . 10 |- (x e. RR -> ({1} X. {x}):{1}-->{x})
10		snssi 3129	. . . . . . . . . 10 |- (x e. RR -> {x} C_ RR)
11		fss 4571	. . . . . . . . . 10 |- ((({1} X. {x}):{1}-->{x} /\ {x} C_ RR) -> ({1} X. {x}):{1}-->RR)
12	9, 10, 11	syl11anc 524	. . . . . . . . 9 |- (x e. RR -> ({1} X. {x}):{1}-->RR)
13		reex 6465	. . . . . . . . . 10 |- RR e. _V
14		snex 3492	. . . . . . . . . 10 |- {1} e. _V
15	13, 14	elmap 5393	. . . . . . . . 9 |- (({1} X. {x}) e. (RR ^m {1}) <-> ({1} X. {x}):{1}-->RR)
16	12, 15	sylibr 217	. . . . . . . 8 |- (x e. RR -> ({1} X. {x}) e. (RR ^m {1}))
17		1z 7368	. . . . . . . . . 10 |- 1 e. ZZ
18		fzsn 7684	. . . . . . . . . 10 |- (1 e. ZZ -> (1...1) = {1})
19	17, 18	ax-mp 7	. . . . . . . . 9 |- (1...1) = {1}
20	19	opreq2i 4893	. . . . . . . 8 |- (RR ^m (1...1)) = (RR ^m {1})
21	16, 20	syl6eleqr 1982	. . . . . . 7 |- (x e. RR -> ({1} X. {x}) e. (RR ^m (1...1)))
22	8, 21	fopab 4800	. . . . . 6 |- {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}:RR-->(RR ^m (1...1))
23		frn 4569	. . . . . 6 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}:RR-->(RR ^m (1...1)) -> ran {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} C_ (RR ^m (1...1)))
24	22, 23	ax-mp 7	. . . . 5 |- ran {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} C_ (RR ^m (1...1))
25	7, 24	sstri 2626	. . . 4 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) C_ (RR ^m (1...1))
26	25	a1i 8	. . 3 |- (Y C_ RR -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) C_ (RR ^m (1...1)))
27	5, 6, 26	sylancr 526	. 2 |- (Y C_ RR -> ((Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp <-> (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) e. (Clsd` (Open` (RRn` 1))) /\ ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Bnd)))
28		hmph 10241	. . . . . 6 |- ((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top) -> (T ~= (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) <-> E.f f e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))))
29	13	opabex2 4539	. . . . . . . 8 |- {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. _V
30		resexg 4250	. . . . . . . 8 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. _V -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. _V)
31	29, 30	ax-mp 7	. . . . . . 7 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. _V
32		eleq1 1957	. . . . . . 7 |- (f = ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) -> (f e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) <-> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))))
33	31, 32	cla4ev 2371	. . . . . 6 |- (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) -> E.f f e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))))
34	28, 33	syl5bir 227	. . . . 5 |- ((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top) -> (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) -> T ~= (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))))
35	34	imp 377	. . . 4 |- (((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top) /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))) -> T ~= (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))
36		comptoppr 10332	. . . . 5 |- ((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top /\ T ~= (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) -> (T e. Comp <-> (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp))
37	36	3expa 1067	. . . 4 |- (((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top) /\ T ~= (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) -> (T e. Comp <-> (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp))
38	35, 37	syldan 516	. . 3 |- (((T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top) /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))) -> (T e. Comp <-> (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp))
39		reheibor.2	. . . . . 6 |- M = (R |` (Y X. Y))
40		reheibor.1	. . . . . . . 8 |- R = ((abs o. - ) |` (RR X. RR))
41	40	remet 9188	. . . . . . 7 |- R e. Met
42		metres 9100	. . . . . . 7 |- (R e. Met -> (R |` (Y X. Y)) e. Met)
43	41, 42	ax-mp 7	. . . . . 6 |- (R |` (Y X. Y)) e. Met
44	39, 43	eqeltri 1967	. . . . 5 |- M e. Met
45		reheibor.3	. . . . . 6 |- T = (Open` M)
46	45	opntop 9147	. . . . 5 |- (M e. Met -> T e. Top)
47	44, 46	ax-mp 7	. . . 4 |- T e. Top
48		rrnmet 16016	. . . . . . 7 |- (1 e. NN -> (RRn` 1) e. Met)
49	6, 48	ax-mp 7	. . . . . 6 |- (RRn` 1) e. Met
50		metres 9100	. . . . . 6 |- ((RRn` 1) e. Met -> ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met)
51	49, 50	ax-mp 7	. . . . 5 |- ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met
52	3	opntop 9147	. . . . 5 |- (((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met -> (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top)
53	51, 52	ax-mp 7	. . . 4 |- (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top
54	47, 53	pm3.2i 307	. . 3 |- (T e. Top /\ (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Top)
55	41	a1i 8	. . . . 5 |- (Y C_ RR -> R e. Met)
56	49	a1i 8	. . . . 5 |- (Y C_ RR -> (RRn` 1) e. Met)
57	40, 8	ismrer1 16024	. . . . . 6 |- {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (RIsmty(RRn` 1))
58	57	a1i 8	. . . . 5 |- (Y C_ RR -> {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (RIsmty(RRn` 1)))
59		id 73	. . . . 5 |- (Y C_ RR -> Y C_ RR)
60	40	remetba 9187	. . . . . 6 |- RR = dom dom R
61		eqid 1884	. . . . . 6 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) = ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)
62	60, 61, 39, 2	ismtyres 15954	. . . . 5 |- (((R e. Met /\ (RRn` 1) e. Met) /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (RIsmty(RRn` 1)) /\ Y C_ RR)) -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (MIsmty((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))
63	55, 56, 58, 59, 62	syl22anc 1101	. . . 4 |- (Y C_ RR -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (MIsmty((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))
64	45, 3	ismtyhmeo 15951	. . . . 5 |- ((M e. Met /\ ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met) -> (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (MIsmty((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))))))
65	44, 51, 64	mp2an 761	. . . 4 |- (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (MIsmty((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))))
66	63, 65	syl 12	. . 3 |- (Y C_ RR -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (T Homeo (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))))
67	38, 54, 66	sylancr 526	. 2 |- (Y C_ RR -> (T e. Comp <-> (Open` ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y)))) e. Comp))
68		reheibor.4	. . . . . . 7 |- U = (Open` R)
69	68	opntop 9147	. . . . . 6 |- (R e. Met -> U e. Top)
70	41, 69	ax-mp 7	. . . . 5 |- U e. Top
71	70	a1i 8	. . . 4 |- (Y C_ RR -> U e. Top)
72	4	opntop 9147	. . . . . 6 |- ((RRn` 1) e. Met -> (Open` (RRn` 1)) e. Top)
73	49, 72	ax-mp 7	. . . . 5 |- (Open` (RRn` 1)) e. Top
74	73	a1i 8	. . . 4 |- (Y C_ RR -> (Open` (RRn` 1)) e. Top)
75	68, 4	ismtyhmeo 15951	. . . . . . 7 |- ((R e. Met /\ (RRn` 1) e. Met) -> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (RIsmty(RRn` 1)) -> {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (U Homeo (Open` (RRn` 1)))))
76	41, 49, 75	mp2an 761	. . . . . 6 |- ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (RIsmty(RRn` 1)) -> {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (U Homeo (Open` (RRn` 1))))
77	57, 76	ax-mp 7	. . . . 5 |- {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (U Homeo (Open` (RRn` 1)))
78	77	a1i 8	. . . 4 |- (Y C_ RR -> {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (U Homeo (Open` (RRn` 1))))
79	60, 68	uniopn2 9138	. . . . . . 7 |- (R e. Met -> U.U = RR)
80	41, 79	ax-mp 7	. . . . . 6 |- U.U = RR
81	80	eqcomi 1888	. . . . 5 |- RR = U.U
82	81	hmeocld 15900	. . . 4 |- (((U e. Top /\ (Open` (RRn` 1)) e. Top) /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} e. (U Homeo (Open` (RRn` 1))) /\ Y C_ RR)) -> (Y e. (Clsd` U) <-> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) e. (Clsd` (Open` (RRn` 1)))))
83	71, 74, 78, 59, 82	syl22anc 1101	. . 3 |- (Y C_ RR -> (Y e. (Clsd` U) <-> ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) e. (Clsd` (Open` (RRn` 1)))))
84	44	a1i 8	. . . 4 |- (Y C_ RR -> M e. Met)
85	51	a1i 8	. . . 4 |- (Y C_ RR -> ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met)
86		ismtybnd 15953	. . . 4 |- ((M e. Met /\ ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Met /\ ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} |` Y) e. (MIsmty((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))))) -> (M e. Bnd <-> ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Bnd))
87	84, 85, 63, 86	syl111anc 1100	. . 3 |- (Y C_ RR -> (M e. Bnd <-> ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Bnd))
88	83, 87	anbi12d 690	. 2 |- (Y C_ RR -> ((Y e. (Clsd` U) /\ M e. Bnd) <-> (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) e. (Clsd` (Open` (RRn` 1))) /\ ((RRn` 1) |` (({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y) X. ({<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}"Y))) e. Bnd)))
89	27, 67, 88	3bitr4d 609	1 |- (Y C_ RR -> (T e. Comp <-> (Y e. (Clsd` U) /\ M e. Bnd)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   C_ wss 2593  {csn 3044  U.cuni 3177   class class class wbr 3338  {copab 3395   X. cxp 3984  ran crn 3987   |` cres 3988  "cima 3989   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  RRcr 6385  1c1 6387   - cmin 6445  NNcn 6449  ZZcz 6451  ...cfz 7637  abscabs 8000  Topctop 8857  Clsdccld 8936  Metcme 9066  Opencopn 9069   Homeo chomeosm 10230   ~= chomeo 10231  Compccomp 10328  Bndcbnd 15931  Ismtycismty 15945  RRncrrn 16011

This theorem is referenced by:  icccmp 16027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-nei 8989  df-lp 9017  df-cn 9030  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-cau 9201  df-cmet 9202  df-homeo 10232  df-hmph 10233  df-comp 10329  df-totbnd 15932  df-bnd 15938  df-ismty 15946  df-rrn 16012

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