Theorem reheibor 33638

Description: Heine-Borel theorem for real numbers. A subset of RR is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
reheibor.2	|- M = ( ( abs o. - ) |` ( Y X. Y ) )
reheibor.3	|- T = ( MetOpen ` M )
reheibor.4	|- U = ( topGen ` ran (,) )

Assertion

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

Proof of Theorem reheibor

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 7572	. . . 4 |- 1o = { (/) }
2		snfi 8038	. . . 4 |- { (/) } e. Fin
3	1, 2	eqeltri 2697	. . 3 |- 1o e. Fin
4		imassrn 5477	. . . . 5 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) C_ ran ( x e. RR |-> ( { (/) } X. { x } ) )
5		0ex 4790	. . . . . . . . . 10 |- (/) e. _V
6		eqid 2622	. . . . . . . . . . 11 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
7		eqid 2622	. . . . . . . . . . 11 |- ( x e. RR |-> ( { (/) } X. { x } ) ) = ( x e. RR |-> ( { (/) } X. { x } ) )
8	6, 7	ismrer1 33637	. . . . . . . . . 10 |- ( (/) e. _V -> ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` { (/) } ) ) )
9	5, 8	ax-mp 5	. . . . . . . . 9 |- ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` { (/) } ) )
10	1	fveq2i 6194	. . . . . . . . . 10 |- ( Rn ` 1o ) = ( Rn ` { (/) } )
11	10	oveq2i 6661	. . . . . . . . 9 |- ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` { (/) } ) )
12	9, 11	eleqtrri 2700	. . . . . . . 8 |- ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) )
13	6	rexmet 22594	. . . . . . . . 9 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
14		eqid 2622	. . . . . . . . . . 11 |- ( RR ^m 1o ) = ( RR ^m 1o )
15	14	rrnmet 33628	. . . . . . . . . 10 |- ( 1o e. Fin -> ( Rn ` 1o ) e. ( Met ` ( RR ^m 1o ) ) )
16		metxmet 22139	. . . . . . . . . 10 |- ( ( Rn ` 1o ) e. ( Met ` ( RR ^m 1o ) ) -> ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) )
17	3, 15, 16	mp2b 10	. . . . . . . . 9 |- ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) )
18		isismty 33600	. . . . . . . . 9 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) ) -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) <-> ( ( x e. RR |-> ( { (/) } X. { x } ) ) : RR -1-1-onto-> ( RR ^m 1o ) /\ A. y e. RR A. z e. RR ( y ( ( abs o. - ) |` ( RR X. RR ) ) z ) = ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` y ) ( Rn ` 1o ) ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` z ) ) ) ) )
19	13, 17, 18	mp2an 708	. . . . . . . 8 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) <-> ( ( x e. RR |-> ( { (/) } X. { x } ) ) : RR -1-1-onto-> ( RR ^m 1o ) /\ A. y e. RR A. z e. RR ( y ( ( abs o. - ) |` ( RR X. RR ) ) z ) = ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` y ) ( Rn ` 1o ) ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` z ) ) ) )
20	12, 19	mpbi 220	. . . . . . 7 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) : RR -1-1-onto-> ( RR ^m 1o ) /\ A. y e. RR A. z e. RR ( y ( ( abs o. - ) |` ( RR X. RR ) ) z ) = ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` y ) ( Rn ` 1o ) ( ( x e. RR |-> ( { (/) } X. { x } ) ) ` z ) ) )
21	20	simpli 474	. . . . . 6 |- ( x e. RR |-> ( { (/) } X. { x } ) ) : RR -1-1-onto-> ( RR ^m 1o )
22		f1of 6137	. . . . . 6 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) : RR -1-1-onto-> ( RR ^m 1o ) -> ( x e. RR |-> ( { (/) } X. { x } ) ) : RR --> ( RR ^m 1o ) )
23		frn 6053	. . . . . 6 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) : RR --> ( RR ^m 1o ) -> ran ( x e. RR |-> ( { (/) } X. { x } ) ) C_ ( RR ^m 1o ) )
24	21, 22, 23	mp2b 10	. . . . 5 |- ran ( x e. RR |-> ( { (/) } X. { x } ) ) C_ ( RR ^m 1o )
25	4, 24	sstri 3612	. . . 4 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) C_ ( RR ^m 1o )
26	25	a1i 11	. . 3 |- ( Y C_ RR -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) C_ ( RR ^m 1o ) )
27		eqid 2622	. . . 4 |- ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) = ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) )
28		eqid 2622	. . . 4 |- ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) = ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) )
29		eqid 2622	. . . 4 |- ( MetOpen ` ( Rn ` 1o ) ) = ( MetOpen ` ( Rn ` 1o ) )
30	14, 27, 28, 29	rrnheibor 33636	. . 3 |- ( ( 1o e. Fin /\ ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) C_ ( RR ^m 1o ) ) -> ( ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp <-> ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) e. ( Clsd ` ( MetOpen ` ( Rn ` 1o ) ) ) /\ ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( Bnd ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
31	3, 26, 30	sylancr 695	. 2 |- ( Y C_ RR -> ( ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp <-> ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) e. ( Clsd ` ( MetOpen ` ( Rn ` 1o ) ) ) /\ ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( Bnd ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
32		reheibor.2	. . . . . . 7 |- M = ( ( abs o. - ) |` ( Y X. Y ) )
33		cnxmet 22576	. . . . . . . 8 |- ( abs o. - ) e. ( *Met ` CC )
34		id 22	. . . . . . . . 9 |- ( Y C_ RR -> Y C_ RR )
35		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
36	34, 35	syl6ss 3615	. . . . . . . 8 |- ( Y C_ RR -> Y C_ CC )
37		xmetres2 22166	. . . . . . . 8 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ Y C_ CC ) -> ( ( abs o. - ) |` ( Y X. Y ) ) e. ( *Met ` Y ) )
38	33, 36, 37	sylancr 695	. . . . . . 7 |- ( Y C_ RR -> ( ( abs o. - ) |` ( Y X. Y ) ) e. ( *Met ` Y ) )
39	32, 38	syl5eqel 2705	. . . . . 6 |- ( Y C_ RR -> M e. ( *Met ` Y ) )
40		xmetres2 22166	. . . . . . 7 |- ( ( ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) /\ ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) C_ ( RR ^m 1o ) ) -> ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( *Met ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) )
41	17, 26, 40	sylancr 695	. . . . . 6 |- ( Y C_ RR -> ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( *Met ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) )
42		reheibor.3	. . . . . . 7 |- T = ( MetOpen ` M )
43	42, 28	ismtyhmeo 33604	. . . . . 6 |- ( ( M e. ( *Met ` Y ) /\ ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( *Met ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) -> ( M Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) C_ ( T Homeo ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ) )
44	39, 41, 43	syl2anc 693	. . . . 5 |- ( Y C_ RR -> ( M Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) C_ ( T Homeo ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ) )
45	13	a1i 11	. . . . . . 7 |- ( Y C_ RR -> ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) )
46	17	a1i 11	. . . . . . 7 |- ( Y C_ RR -> ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) )
47	12	a1i 11	. . . . . . 7 |- ( Y C_ RR -> ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) )
48		eqid 2622	. . . . . . . 8 |- ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) = ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y )
49		eqid 2622	. . . . . . . 8 |- ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) = ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) )
50	48, 49, 27	ismtyres 33607	. . . . . . 7 |- ( ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) ) /\ ( ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) /\ Y C_ RR ) ) -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
51	45, 46, 47, 34, 50	syl22anc 1327	. . . . . 6 |- ( Y C_ RR -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
52		xpss12 5225	. . . . . . . . . 10 |- ( ( Y C_ RR /\ Y C_ RR ) -> ( Y X. Y ) C_ ( RR X. RR ) )
53	52	anidms 677	. . . . . . . . 9 |- ( Y C_ RR -> ( Y X. Y ) C_ ( RR X. RR ) )
54	53	resabs1d 5428	. . . . . . . 8 |- ( Y C_ RR -> ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) = ( ( abs o. - ) |` ( Y X. Y ) ) )
55	54, 32	syl6eqr 2674	. . . . . . 7 |- ( Y C_ RR -> ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) = M )
56	55	oveq1d 6665	. . . . . 6 |- ( Y C_ RR -> ( ( ( ( abs o. - ) |` ( RR X. RR ) ) |` ( Y X. Y ) ) Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) = ( M Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
57	51, 56	eleqtrd 2703	. . . . 5 |- ( Y C_ RR -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( M Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
58	44, 57	sseldd 3604	. . . 4 |- ( Y C_ RR -> ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( T Homeo ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ) )
59		hmphi 21580	. . . 4 |- ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( T Homeo ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ) -> T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
60	58, 59	syl 17	. . 3 |- ( Y C_ RR -> T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
61		cmphmph 21591	. . . 4 |- ( T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) -> ( T e. Comp -> ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp ) )
62		hmphsym 21585	. . . . 5 |- ( T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) -> ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ~= T )
63		cmphmph 21591	. . . . 5 |- ( ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ~= T -> ( ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp -> T e. Comp ) )
64	62, 63	syl 17	. . . 4 |- ( T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) -> ( ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp -> T e. Comp ) )
65	61, 64	impbid 202	. . 3 |- ( T ~= ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) -> ( T e. Comp <-> ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp ) )
66	60, 65	syl 17	. 2 |- ( Y C_ RR -> ( T e. Comp <-> ( MetOpen ` ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) e. Comp ) )
67		reheibor.4	. . . . . . . 8 |- U = ( topGen ` ran (,) )
68		eqid 2622	. . . . . . . . 9 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
69	6, 68	tgioo 22599	. . . . . . . 8 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
70	67, 69	eqtri 2644	. . . . . . 7 |- U = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
71	70, 29	ismtyhmeo 33604	. . . . . 6 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ ( Rn ` 1o ) e. ( *Met ` ( RR ^m 1o ) ) ) -> ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) C_ ( U Homeo ( MetOpen ` ( Rn ` 1o ) ) ) )
72	13, 17, 71	mp2an 708	. . . . 5 |- ( ( ( abs o. - ) |` ( RR X. RR ) ) Ismty ( Rn ` 1o ) ) C_ ( U Homeo ( MetOpen ` ( Rn ` 1o ) ) )
73	72, 12	sselii 3600	. . . 4 |- ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( U Homeo ( MetOpen ` ( Rn ` 1o ) ) )
74		retopon 22567	. . . . . . 7 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
75	67, 74	eqeltri 2697	. . . . . 6 |- U e. (TopOn ` RR )
76	75	toponunii 20721	. . . . 5 |- RR = U. U
77	76	hmeocld 21570	. . . 4 |- ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) e. ( U Homeo ( MetOpen ` ( Rn ` 1o ) ) ) /\ Y C_ RR ) -> ( Y e. ( Clsd ` U ) <-> ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) e. ( Clsd ` ( MetOpen ` ( Rn ` 1o ) ) ) ) )
78	73, 34, 77	sylancr 695	. . 3 |- ( Y C_ RR -> ( Y e. ( Clsd ` U ) <-> ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) e. ( Clsd ` ( MetOpen ` ( Rn ` 1o ) ) ) ) )
79		ismtybnd 33606	. . . 4 |- ( ( M e. ( *Met ` Y ) /\ ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( *Met ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) /\ ( ( x e. RR |-> ( { (/) } X. { x } ) ) |` Y ) e. ( M Ismty ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) ) -> ( M e. ( Bnd ` Y ) <-> ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( Bnd ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) )
80	39, 41, 57, 79	syl3anc 1326	. . 3 |- ( Y C_ RR -> ( M e. ( Bnd ` Y ) <-> ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( Bnd ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) )
81	78, 80	anbi12d 747	. 2 |- ( Y C_ RR -> ( ( Y e. ( Clsd ` U ) /\ M e. ( Bnd ` Y ) ) <-> ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) e. ( Clsd ` ( MetOpen ` ( Rn ` 1o ) ) ) /\ ( ( Rn ` 1o ) |` ( ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) X. ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) e. ( Bnd ` ( ( x e. RR |-> ( { (/) } X. { x } ) ) " Y ) ) ) ) )
82	31, 66, 81	3bitr4d 300	1 |- ( Y C_ RR -> ( T e. Comp <-> ( Y e. ( Clsd ` U ) /\ M e. ( Bnd ` Y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   - cmin 10266   (,)cioo 12175   abscabs 13974   topGenctg 16098   *Metcxmt 19731   Metcme 19732   MetOpencmopn 19736  TopOnctopon 20715   Clsdccld 20820   Compccmp 21189   Homeochmeo 21556   ~= chmph 21557   Bndcbnd 33566   Ismty cismty 33597   Rncrrn 33624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-topgen 16104  df-prds 16108  df-pws 16110  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-lm 21033  df-haus 21119  df-cmp 21190  df-hmeo 21558  df-hmph 21559  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-cfil 23053  df-cau 23054  df-cmet 23055  df-totbnd 33567  df-bnd 33578  df-ismty 33598  df-rrn 33625

This theorem is referenced by:  icccmpALT  33640