Theorem dvcnvrelem2 23781

Description: Lemma for dvcnvre 23782. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Hypotheses

Ref	Expression
dvcnvre.f	|- ( ph -> F e. ( X -cn-> RR ) )
dvcnvre.d	|- ( ph -> dom ( RR _D F ) = X )
dvcnvre.z	|- ( ph -> -. 0 e. ran ( RR _D F ) )
dvcnvre.1	|- ( ph -> F : X -1-1-onto-> Y )
dvcnvre.c	|- ( ph -> C e. X )
dvcnvre.r	|- ( ph -> R e. RR+ )
dvcnvre.s	|- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ X )
dvcnvre.t	|- T = ( topGen ` ran (,) )
dvcnvre.j	|- J = ( TopOpen ` ℂfld )
dvcnvre.m	|- M = ( J ↾t X )
dvcnvre.n	|- N = ( J ↾t Y )

Assertion

Ref	Expression
dvcnvrelem2	|- ( ph -> ( ( F ` C ) e. ( ( int ` T ) ` Y ) /\ `' F e. ( ( N CnP M ) ` ( F ` C ) ) ) )

Proof of Theorem dvcnvrelem2

Step	Hyp	Ref	Expression
1		dvcnvre.t	. . . . . 6 |- T = ( topGen ` ran (,) )
2		retop 22565	. . . . . 6 |- ( topGen ` ran (,) ) e. Top
3	1, 2	eqeltri 2697	. . . . 5 |- T e. Top
4	3	a1i 11	. . . 4 |- ( ph -> T e. Top )
5		dvcnvre.1	. . . . . 6 |- ( ph -> F : X -1-1-onto-> Y )
6		f1ofo 6144	. . . . . 6 |- ( F : X -1-1-onto-> Y -> F : X -onto-> Y )
7		forn 6118	. . . . . 6 |- ( F : X -onto-> Y -> ran F = Y )
8	5, 6, 7	3syl 18	. . . . 5 |- ( ph -> ran F = Y )
9		dvcnvre.f	. . . . . 6 |- ( ph -> F e. ( X -cn-> RR ) )
10		cncff 22696	. . . . . 6 |- ( F e. ( X -cn-> RR ) -> F : X --> RR )
11		frn 6053	. . . . . 6 |- ( F : X --> RR -> ran F C_ RR )
12	9, 10, 11	3syl 18	. . . . 5 |- ( ph -> ran F C_ RR )
13	8, 12	eqsstr3d 3640	. . . 4 |- ( ph -> Y C_ RR )
14		imassrn 5477	. . . . 5 |- ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ ran F
15	14, 8	syl5sseq 3653	. . . 4 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ Y )
16		uniretop 22566	. . . . . 6 |- RR = U. ( topGen ` ran (,) )
17	1	unieqi 4445	. . . . . 6 |- U. T = U. ( topGen ` ran (,) )
18	16, 17	eqtr4i 2647	. . . . 5 |- RR = U. T
19	18	ntrss 20859	. . . 4 |- ( ( T e. Top /\ Y C_ RR /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ Y ) -> ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( ( int ` T ) ` Y ) )
20	4, 13, 15, 19	syl3anc 1326	. . 3 |- ( ph -> ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( ( int ` T ) ` Y ) )
21		dvcnvre.d	. . . . 5 |- ( ph -> dom ( RR _D F ) = X )
22		dvcnvre.z	. . . . 5 |- ( ph -> -. 0 e. ran ( RR _D F ) )
23		dvcnvre.c	. . . . 5 |- ( ph -> C e. X )
24		dvcnvre.r	. . . . 5 |- ( ph -> R e. RR+ )
25		dvcnvre.s	. . . . 5 |- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ X )
26	9, 21, 22, 5, 23, 24, 25	dvcnvrelem1 23780	. . . 4 |- ( ph -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
27	1	fveq2i 6194	. . . . 5 |- ( int ` T ) = ( int ` ( topGen ` ran (,) ) )
28	27	fveq1i 6192	. . . 4 |- ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) )
29	26, 28	syl6eleqr 2712	. . 3 |- ( ph -> ( F ` C ) e. ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
30	20, 29	sseldd 3604	. 2 |- ( ph -> ( F ` C ) e. ( ( int ` T ) ` Y ) )
31		f1ocnv 6149	. . . . . . 7 |- ( F : X -1-1-onto-> Y -> `' F : Y -1-1-onto-> X )
32		f1of 6137	. . . . . . 7 |- ( `' F : Y -1-1-onto-> X -> `' F : Y --> X )
33	5, 31, 32	3syl 18	. . . . . 6 |- ( ph -> `' F : Y --> X )
34		ffun 6048	. . . . . 6 |- ( `' F : Y --> X -> Fun `' F )
35		funcnvres 5967	. . . . . 6 |- ( Fun `' F -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
36	33, 34, 35	3syl 18	. . . . 5 |- ( ph -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
37		dvbsss 23666	. . . . . . . . . . 11 |- dom ( RR _D F ) C_ RR
38	21, 37	syl6eqssr 3656	. . . . . . . . . 10 |- ( ph -> X C_ RR )
39		ax-resscn 9993	. . . . . . . . . 10 |- RR C_ CC
40	38, 39	syl6ss 3615	. . . . . . . . 9 |- ( ph -> X C_ CC )
41		cncfss 22702	. . . . . . . . 9 |- ( ( ( ( C - R ) [,] ( C + R ) ) C_ X /\ X C_ CC ) -> ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> ( ( C - R ) [,] ( C + R ) ) ) C_ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> X ) )
42	25, 40, 41	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> ( ( C - R ) [,] ( C + R ) ) ) C_ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> X ) )
43		f1of1 6136	. . . . . . . . . . 11 |- ( F : X -1-1-onto-> Y -> F : X -1-1-> Y )
44	5, 43	syl 17	. . . . . . . . . 10 |- ( ph -> F : X -1-1-> Y )
45		f1ores 6151	. . . . . . . . . 10 |- ( ( F : X -1-1-> Y /\ ( ( C - R ) [,] ( C + R ) ) C_ X ) -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) -1-1-onto-> ( F " ( ( C - R ) [,] ( C + R ) ) ) )
46	44, 25, 45	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) -1-1-onto-> ( F " ( ( C - R ) [,] ( C + R ) ) ) )
47		dvcnvre.j	. . . . . . . . . . . . . . 15 |- J = ( TopOpen ` ℂfld )
48	47	tgioo2 22606	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( J ↾t RR )
49	1, 48	eqtri 2644	. . . . . . . . . . . . 13 |- T = ( J ↾t RR )
50	49	oveq1i 6660	. . . . . . . . . . . 12 |- ( T ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( ( J ↾t RR ) ↾t ( ( C - R ) [,] ( C + R ) ) )
51	47	cnfldtop 22587	. . . . . . . . . . . . . 14 |- J e. Top
52	51	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
53	25, 38	sstrd 3613	. . . . . . . . . . . . 13 |- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ RR )
54		reex 10027	. . . . . . . . . . . . . 14 |- RR e. _V
55	54	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR e. _V )
56		restabs 20969	. . . . . . . . . . . . 13 |- ( ( J e. Top /\ ( ( C - R ) [,] ( C + R ) ) C_ RR /\ RR e. _V ) -> ( ( J ↾t RR ) ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) )
57	52, 53, 55, 56	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( ( J ↾t RR ) ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) )
58	50, 57	syl5eq 2668	. . . . . . . . . . 11 |- ( ph -> ( T ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) )
59	38, 23	sseldd 3604	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
60	24	rpred 11872	. . . . . . . . . . . . 13 |- ( ph -> R e. RR )
61	59, 60	resubcld 10458	. . . . . . . . . . . 12 |- ( ph -> ( C - R ) e. RR )
62	59, 60	readdcld 10069	. . . . . . . . . . . 12 |- ( ph -> ( C + R ) e. RR )
63		eqid 2622	. . . . . . . . . . . . 13 |- ( T ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( T ↾t ( ( C - R ) [,] ( C + R ) ) )
64	1, 63	icccmp 22628	. . . . . . . . . . . 12 |- ( ( ( C - R ) e. RR /\ ( C + R ) e. RR ) -> ( T ↾t ( ( C - R ) [,] ( C + R ) ) ) e. Comp )
65	61, 62, 64	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( T ↾t ( ( C - R ) [,] ( C + R ) ) ) e. Comp )
66	58, 65	eqeltrrd 2702	. . . . . . . . . 10 |- ( ph -> ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) e. Comp )
67		f1of 6137	. . . . . . . . . . . 12 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) -1-1-onto-> ( F " ( ( C - R ) [,] ( C + R ) ) ) -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) --> ( F " ( ( C - R ) [,] ( C + R ) ) ) )
68	46, 67	syl 17	. . . . . . . . . . 11 |- ( ph -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) --> ( F " ( ( C - R ) [,] ( C + R ) ) ) )
69	12, 39	syl6ss 3615	. . . . . . . . . . . . 13 |- ( ph -> ran F C_ CC )
70	14, 69	syl5ss 3614	. . . . . . . . . . . 12 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ CC )
71		rescncf 22700	. . . . . . . . . . . . 13 |- ( ( ( C - R ) [,] ( C + R ) ) C_ X -> ( F e. ( X -cn-> RR ) -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> RR ) ) )
72	25, 9, 71	sylc 65	. . . . . . . . . . . 12 |- ( ph -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> RR ) )
73		cncffvrn 22701	. . . . . . . . . . . 12 |- ( ( ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ CC /\ ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> RR ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) <-> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) --> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
74	70, 72, 73	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) <-> ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) --> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
75	68, 74	mpbird 247	. . . . . . . . . 10 |- ( ph -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
76		eqid 2622	. . . . . . . . . . 11 |- ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) = ( J ↾t ( ( C - R ) [,] ( C + R ) ) )
77	47, 76	cncfcnvcn 22724	. . . . . . . . . 10 |- ( ( ( J ↾t ( ( C - R ) [,] ( C + R ) ) ) e. Comp /\ ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> ( F " ( ( C - R ) [,] ( C + R ) ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) -1-1-onto-> ( F " ( ( C - R ) [,] ( C + R ) ) ) <-> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> ( ( C - R ) [,] ( C + R ) ) ) ) )
78	66, 75, 77	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) : ( ( C - R ) [,] ( C + R ) ) -1-1-onto-> ( F " ( ( C - R ) [,] ( C + R ) ) ) <-> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> ( ( C - R ) [,] ( C + R ) ) ) ) )
79	46, 78	mpbid 222	. . . . . . . 8 |- ( ph -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> ( ( C - R ) [,] ( C + R ) ) ) )
80	42, 79	sseldd 3604	. . . . . . 7 |- ( ph -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> X ) )
81		eqid 2622	. . . . . . . . 9 |- ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) )
82		dvcnvre.m	. . . . . . . . 9 |- M = ( J ↾t X )
83	47, 81, 82	cncfcn 22712	. . . . . . . 8 |- ( ( ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ CC /\ X C_ CC ) -> ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> X ) = ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) Cn M ) )
84	70, 40, 83	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( F " ( ( C - R ) [,] ( C + R ) ) ) -cn-> X ) = ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) Cn M ) )
85	80, 84	eleqtrd 2703	. . . . . 6 |- ( ph -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) Cn M ) )
86	59, 24	ltsubrpd 11904	. . . . . . . . . 10 |- ( ph -> ( C - R ) < C )
87	61, 59, 86	ltled 10185	. . . . . . . . 9 |- ( ph -> ( C - R ) <_ C )
88	59, 24	ltaddrpd 11905	. . . . . . . . . 10 |- ( ph -> C < ( C + R ) )
89	59, 62, 88	ltled 10185	. . . . . . . . 9 |- ( ph -> C <_ ( C + R ) )
90		elicc2 12238	. . . . . . . . . 10 |- ( ( ( C - R ) e. RR /\ ( C + R ) e. RR ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) <-> ( C e. RR /\ ( C - R ) <_ C /\ C <_ ( C + R ) ) ) )
91	61, 62, 90	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( C e. ( ( C - R ) [,] ( C + R ) ) <-> ( C e. RR /\ ( C - R ) <_ C /\ C <_ ( C + R ) ) ) )
92	59, 87, 89, 91	mpbir3and 1245	. . . . . . . 8 |- ( ph -> C e. ( ( C - R ) [,] ( C + R ) ) )
93		ffun 6048	. . . . . . . . . 10 |- ( F : X --> RR -> Fun F )
94	9, 10, 93	3syl 18	. . . . . . . . 9 |- ( ph -> Fun F )
95		fdm 6051	. . . . . . . . . . 11 |- ( F : X --> RR -> dom F = X )
96	9, 10, 95	3syl 18	. . . . . . . . . 10 |- ( ph -> dom F = X )
97	25, 96	sseqtr4d 3642	. . . . . . . . 9 |- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ dom F )
98		funfvima2 6493	. . . . . . . . 9 |- ( ( Fun F /\ ( ( C - R ) [,] ( C + R ) ) C_ dom F ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` C ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
99	94, 97, 98	syl2anc 693	. . . . . . . 8 |- ( ph -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` C ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
100	92, 99	mpd 15	. . . . . . 7 |- ( ph -> ( F ` C ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) )
101	47	cnfldtopon 22586	. . . . . . . . 9 |- J e. (TopOn ` CC )
102		resttopon 20965	. . . . . . . . 9 |- ( ( J e. (TopOn ` CC ) /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ CC ) -> ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. (TopOn ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
103	101, 70, 102	sylancr 695	. . . . . . . 8 |- ( ph -> ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. (TopOn ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
104		toponuni 20719	. . . . . . . 8 |- ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. (TopOn ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F " ( ( C - R ) [,] ( C + R ) ) ) = U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
105	103, 104	syl 17	. . . . . . 7 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) = U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
106	100, 105	eleqtrd 2703	. . . . . 6 |- ( ph -> ( F ` C ) e. U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
107		eqid 2622	. . . . . . 7 |- U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) )
108	107	cncnpi 21082	. . . . . 6 |- ( ( `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) Cn M ) /\ ( F ` C ) e. U. ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) ) -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) )
109	85, 106, 108	syl2anc 693	. . . . 5 |- ( ph -> `' ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) )
110	36, 109	eqeltrrd 2702	. . . 4 |- ( ph -> ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. ( ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) )
111		dvcnvre.n	. . . . . . . 8 |- N = ( J ↾t Y )
112	111	oveq1i 6660	. . . . . . 7 |- ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( J ↾t Y ) ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) )
113		ssexg 4804	. . . . . . . . 9 |- ( ( Y C_ RR /\ RR e. _V ) -> Y e. _V )
114	13, 54, 113	sylancl 694	. . . . . . . 8 |- ( ph -> Y e. _V )
115		restabs 20969	. . . . . . . 8 |- ( ( J e. Top /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ Y /\ Y e. _V ) -> ( ( J ↾t Y ) ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
116	52, 15, 114, 115	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( J ↾t Y ) ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
117	112, 116	syl5eq 2668	. . . . . 6 |- ( ph -> ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
118	117	oveq1d 6665	. . . . 5 |- ( ph -> ( ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) = ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) )
119	118	fveq1d 6193	. . . 4 |- ( ph -> ( ( ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) = ( ( ( J ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) )
120	110, 119	eleqtrrd 2704	. . 3 |- ( ph -> ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. ( ( ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) )
121	13, 39	syl6ss 3615	. . . . . . 7 |- ( ph -> Y C_ CC )
122		resttopon 20965	. . . . . . 7 |- ( ( J e. (TopOn ` CC ) /\ Y C_ CC ) -> ( J ↾t Y ) e. (TopOn ` Y ) )
123	101, 121, 122	sylancr 695	. . . . . 6 |- ( ph -> ( J ↾t Y ) e. (TopOn ` Y ) )
124	111, 123	syl5eqel 2705	. . . . 5 |- ( ph -> N e. (TopOn ` Y ) )
125		topontop 20718	. . . . 5 |- ( N e. (TopOn ` Y ) -> N e. Top )
126	124, 125	syl 17	. . . 4 |- ( ph -> N e. Top )
127		toponuni 20719	. . . . . 6 |- ( N e. (TopOn ` Y ) -> Y = U. N )
128	124, 127	syl 17	. . . . 5 |- ( ph -> Y = U. N )
129	15, 128	sseqtrd 3641	. . . 4 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ U. N )
130	15, 13	sstrd 3613	. . . . . . . . 9 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ RR )
131		difssd 3738	. . . . . . . . 9 |- ( ph -> ( RR \ Y ) C_ RR )
132	130, 131	unssd 3789	. . . . . . . 8 |- ( ph -> ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) C_ RR )
133		ssun1 3776	. . . . . . . . 9 |- ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) )
134	133	a1i 11	. . . . . . . 8 |- ( ph -> ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) )
135	18	ntrss 20859	. . . . . . . 8 |- ( ( T e. Top /\ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) C_ RR /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) -> ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) )
136	4, 132, 134, 135	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( int ` T ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) )
137	136, 29	sseldd 3604	. . . . . 6 |- ( ph -> ( F ` C ) e. ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) )
138		f1of 6137	. . . . . . . 8 |- ( F : X -1-1-onto-> Y -> F : X --> Y )
139	5, 138	syl 17	. . . . . . 7 |- ( ph -> F : X --> Y )
140	139, 23	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( F ` C ) e. Y )
141	137, 140	elind 3798	. . . . 5 |- ( ph -> ( F ` C ) e. ( ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) i^i Y ) )
142		eqid 2622	. . . . . . . 8 |- ( T ↾t Y ) = ( T ↾t Y )
143	18, 142	restntr 20986	. . . . . . 7 |- ( ( T e. Top /\ Y C_ RR /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ Y ) -> ( ( int ` ( T ↾t Y ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) i^i Y ) )
144	4, 13, 15, 143	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( int ` ( T ↾t Y ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) i^i Y ) )
145		restabs 20969	. . . . . . . . . 10 |- ( ( J e. Top /\ Y C_ RR /\ RR e. _V ) -> ( ( J ↾t RR ) ↾t Y ) = ( J ↾t Y ) )
146	52, 13, 55, 145	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( J ↾t RR ) ↾t Y ) = ( J ↾t Y ) )
147	49	oveq1i 6660	. . . . . . . . 9 |- ( T ↾t Y ) = ( ( J ↾t RR ) ↾t Y )
148	146, 147, 111	3eqtr4g 2681	. . . . . . . 8 |- ( ph -> ( T ↾t Y ) = N )
149	148	fveq2d 6195	. . . . . . 7 |- ( ph -> ( int ` ( T ↾t Y ) ) = ( int ` N ) )
150	149	fveq1d 6193	. . . . . 6 |- ( ph -> ( ( int ` ( T ↾t Y ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( int ` N ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
151	144, 150	eqtr3d 2658	. . . . 5 |- ( ph -> ( ( ( int ` T ) ` ( ( F " ( ( C - R ) [,] ( C + R ) ) ) u. ( RR \ Y ) ) ) i^i Y ) = ( ( int ` N ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
152	141, 151	eleqtrd 2703	. . . 4 |- ( ph -> ( F ` C ) e. ( ( int ` N ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
153	128	feq2d 6031	. . . . . 6 |- ( ph -> ( `' F : Y --> X <-> `' F : U. N --> X ) )
154	33, 153	mpbid 222	. . . . 5 |- ( ph -> `' F : U. N --> X )
155		resttopon 20965	. . . . . . . 8 |- ( ( J e. (TopOn ` CC ) /\ X C_ CC ) -> ( J ↾t X ) e. (TopOn ` X ) )
156	101, 40, 155	sylancr 695	. . . . . . 7 |- ( ph -> ( J ↾t X ) e. (TopOn ` X ) )
157	82, 156	syl5eqel 2705	. . . . . 6 |- ( ph -> M e. (TopOn ` X ) )
158		toponuni 20719	. . . . . 6 |- ( M e. (TopOn ` X ) -> X = U. M )
159		feq3 6028	. . . . . 6 |- ( X = U. M -> ( `' F : U. N --> X <-> `' F : U. N --> U. M ) )
160	157, 158, 159	3syl 18	. . . . 5 |- ( ph -> ( `' F : U. N --> X <-> `' F : U. N --> U. M ) )
161	154, 160	mpbid 222	. . . 4 |- ( ph -> `' F : U. N --> U. M )
162		eqid 2622	. . . . 5 |- U. N = U. N
163		eqid 2622	. . . . 5 |- U. M = U. M
164	162, 163	cnprest 21093	. . . 4 |- ( ( ( N e. Top /\ ( F " ( ( C - R ) [,] ( C + R ) ) ) C_ U. N ) /\ ( ( F ` C ) e. ( ( int ` N ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) /\ `' F : U. N --> U. M ) ) -> ( `' F e. ( ( N CnP M ) ` ( F ` C ) ) <-> ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. ( ( ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) ) )
165	126, 129, 152, 161, 164	syl22anc 1327	. . 3 |- ( ph -> ( `' F e. ( ( N CnP M ) ` ( F ` C ) ) <-> ( `' F |` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) e. ( ( ( N ↾t ( F " ( ( C - R ) [,] ( C + R ) ) ) ) CnP M ) ` ( F ` C ) ) ) )
166	120, 165	mpbird 247	. 2 |- ( ph -> `' F e. ( ( N CnP M ) ` ( F ` C ) ) )
167	30, 166	jca 554	1 |- ( ph -> ( ( F ` C ) e. ( ( int ` T ) ` Y ) /\ `' F e. ( ( N CnP M ) ` ( F ` C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   <_ cle 10075   - cmin 10266   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   Compccmp 21189   -cn->ccncf 22679   _D cdv 23627

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-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-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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvcnvre  23782