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Theorem dvcnvrelem2 19894
Description: Lemma for dvcnvre 19895. (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 ` fld )
dvcnvre.m  |-  M  =  ( Jt  X )
dvcnvre.n  |-  N  =  ( Jt  Y )
Assertion
Ref Expression
dvcnvrelem2  |-  ( ph  ->  ( ( F `  C )  e.  ( ( int `  T
) `  Y )  /\  `' F  e.  (
( N  CnP  M
) `  ( F `  C ) ) ) )

Proof of Theorem dvcnvrelem2
StepHypRef Expression
1 dvcnvre.t . . . . . 6  |-  T  =  ( topGen `  ran  (,) )
2 retop 18787 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
31, 2eqeltri 2505 . . . . 5  |-  T  e. 
Top
43a1i 11 . . . 4  |-  ( ph  ->  T  e.  Top )
5 dvcnvre.1 . . . . . 6  |-  ( ph  ->  F : X -1-1-onto-> Y )
6 f1ofo 5673 . . . . . 6  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
7 forn 5648 . . . . . 6  |-  ( F : X -onto-> Y  ->  ran  F  =  Y )
85, 6, 73syl 19 . . . . 5  |-  ( ph  ->  ran  F  =  Y )
9 dvcnvre.f . . . . . 6  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
10 cncff 18915 . . . . . 6  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
11 frn 5589 . . . . . 6  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
129, 10, 113syl 19 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
138, 12eqsstr3d 3375 . . . 4  |-  ( ph  ->  Y  C_  RR )
14 imassrn 5208 . . . . 5  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ran  F
1514, 8syl5sseq 3388 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  Y )
16 uniretop 18788 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
171unieqi 4017 . . . . . 6  |-  U. T  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2458 . . . . 5  |-  RR  =  U. T
1918ntrss 17111 . . . 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 )
)
204, 13, 15, 19syl3anc 1184 . . 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 )
269, 21, 22, 5, 23, 24, 25dvcnvrelem1 19893 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
271fveq2i 5723 . . . . 5  |-  ( int `  T )  =  ( int `  ( topGen ` 
ran  (,) ) )
2827fveq1i 5721 . . . 4  |-  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )
2926, 28syl6eleqr 2526 . . 3  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
3020, 29sseldd 3341 . 2  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 Y ) )
31 f1ocnv 5679 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
32 f1of 5666 . . . . . . 7  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
335, 31, 323syl 19 . . . . . 6  |-  ( ph  ->  `' F : Y --> X )
34 ffun 5585 . . . . . 6  |-  ( `' F : Y --> X  ->  Fun  `' F )
35 funcnvres 5514 . . . . . 6  |-  ( Fun  `' F  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
3633, 34, 353syl 19 . . . . 5  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( `' F  |`  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
37 dvbsss 19781 . . . . . . . . . . 11  |-  dom  ( RR  _D  F )  C_  RR
3821, 37syl6eqssr 3391 . . . . . . . . . 10  |-  ( ph  ->  X  C_  RR )
39 ax-resscn 9039 . . . . . . . . . 10  |-  RR  C_  CC
4038, 39syl6ss 3352 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
41 cncfss 18921 . . . . . . . . 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 ) )
4225, 40, 41syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) -cn-> ( ( C  -  R ) [,] ( C  +  R ) ) ) 
C_  ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) -cn-> X ) )
43 f1of1 5665 . . . . . . . . . . 11  |-  ( F : X -1-1-onto-> Y  ->  F : X -1-1-> Y )
445, 43syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : X -1-1-> Y
)
45 f1ores 5681 . . . . . . . . . 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
) ) ) )
4644, 25, 45syl2anc 643 . . . . . . . . 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 ` fld )
4847tgioo2 18826 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( Jt  RR )
491, 48eqtri 2455 . . . . . . . . . . . . 13  |-  T  =  ( Jt  RR )
5049oveq1i 6083 . . . . . . . . . . . 12  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( ( Jt  RR )t  ( ( C  -  R ) [,] ( C  +  R
) ) )
5147cnfldtop 18810 . . . . . . . . . . . . . 14  |-  J  e. 
Top
5251a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
5325, 38sstrd 3350 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
54 reex 9073 . . . . . . . . . . . . . 14  |-  RR  e.  _V
5554a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  e.  _V )
56 restabs 17221 . . . . . . . . . . . . 13  |-  ( ( J  e.  Top  /\  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR  /\  RR  e.  _V )  ->  (
( Jt  RR )t  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) ) )
5752, 53, 55, 56syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( Jt  RR )t  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) ) )
5850, 57syl5eq 2479 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) ) )
5938, 23sseldd 3341 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
6024rpred 10640 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
6159, 60resubcld 9457 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR )
6259, 60readdcld 9107 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR )
63 eqid 2435 . . . . . . . . . . . . 13  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )
641, 63icccmp 18848 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R ) ) )  e.  Comp )
6561, 62, 64syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
6658, 65eqeltrrd 2510 . . . . . . . . . 10  |-  ( ph  ->  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
67 f1of 5666 . . . . . . . . . . . 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 ) ) ) )
6846, 67syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) : ( ( C  -  R ) [,] ( C  +  R ) ) --> ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
6912, 39syl6ss 3352 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  F  C_  CC )
7014, 69syl5ss 3351 . . . . . . . . . . . 12  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  CC )
71 rescncf 18919 . . . . . . . . . . . . 13  |-  ( ( ( C  -  R
) [,] ( C  +  R ) ) 
C_  X  ->  ( F  e.  ( X -cn->
RR )  ->  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  e.  ( ( ( C  -  R ) [,] ( C  +  R
) ) -cn-> RR ) ) )
7225, 9, 71sylc 58 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> RR ) )
73 cncffvrn 18920 . . . . . . . . . . . 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 ) ) ) ) )
7470, 72, 73syl2anc 643 . . . . . . . . . . 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 ) ) ) ) )
7568, 74mpbird 224 . . . . . . . . . 10  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) ) )
76 eqid 2435 . . . . . . . . . . 11  |-  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )
7747, 76cncfcnvcn 18943 . . . . . . . . . 10  |-  ( ( ( Jt  ( ( 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
) ) ) ) )
7866, 75, 77syl2anc 643 . . . . . . . . 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
) ) ) ) )
7946, 78mpbid 202 . . . . . . . 8  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> ( ( C  -  R ) [,] ( C  +  R
) ) ) )
8042, 79sseldd 3341 . . . . . . 7  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> X ) )
81 eqid 2435 . . . . . . . . 9  |-  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
82 dvcnvre.m . . . . . . . . 9  |-  M  =  ( Jt  X )
8347, 81, 82cncfcn 18931 . . . . . . . 8  |-  ( ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  CC  /\  X  C_  CC )  ->  (
( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> X )  =  ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  Cn  M ) )
8470, 40, 83syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) -cn-> X )  =  ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  Cn  M ) )
8580, 84eleqtrd 2511 . . . . . 6  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  Cn  M ) )
8659, 24ltsubrpd 10668 . . . . . . . . . 10  |-  ( ph  ->  ( C  -  R
)  <  C )
8761, 59, 86ltled 9213 . . . . . . . . 9  |-  ( ph  ->  ( C  -  R
)  <_  C )
8859, 24ltaddrpd 10669 . . . . . . . . . 10  |-  ( ph  ->  C  <  ( C  +  R ) )
8959, 62, 88ltled 9213 . . . . . . . . 9  |-  ( ph  ->  C  <_  ( C  +  R ) )
90 elicc2 10967 . . . . . . . . . 10  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  <->  ( C  e.  RR  /\  ( C  -  R )  <_  C  /\  C  <_  ( C  +  R )
) ) )
9161, 62, 90syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  <-> 
( C  e.  RR  /\  ( C  -  R
)  <_  C  /\  C  <_  ( C  +  R ) ) ) )
9259, 87, 89, 91mpbir3and 1137 . . . . . . . 8  |-  ( ph  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
93 ffun 5585 . . . . . . . . . 10  |-  ( F : X --> RR  ->  Fun 
F )
949, 10, 933syl 19 . . . . . . . . 9  |-  ( ph  ->  Fun  F )
95 fdm 5587 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  dom 
F  =  X )
969, 10, 953syl 19 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  X )
9725, 96sseqtr4d 3377 . . . . . . . . 9  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  dom  F )
98 funfvima2 5966 . . . . . . . . 9  |-  ( ( Fun  F  /\  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  C )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
9994, 97, 98syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  C )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
10092, 99mpd 15 . . . . . . 7  |-  ( ph  ->  ( F `  C
)  e.  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )
10147cnfldtopon 18809 . . . . . . . . 9  |-  J  e.  (TopOn `  CC )
102 resttopon 17217 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  CC )  /\  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  CC )  ->  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  (TopOn `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
103101, 70, 102sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  e.  (TopOn `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) ) )
104 toponuni 16984 . . . . . . . 8  |-  ( ( Jt  ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) )  e.  (TopOn `  ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) )  ->  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  = 
U. ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
105103, 104syl 16 . . . . . . 7  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
106100, 105eleqtrd 2511 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
107 eqid 2435 . . . . . . 7  |-  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
108107cncnpi 17334 . . . . . 6  |-  ( ( `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  Cn  M )  /\  ( F `  C )  e.  U. ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  e.  ( ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) ) )
10985, 106, 108syl2anc 643 . . . . 5  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) `  ( F `  C ) ) )
11036, 109eqeltrrd 2510 . . . 4  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
111 dvcnvre.n . . . . . . . 8  |-  N  =  ( Jt  Y )
112111oveq1i 6083 . . . . . . 7  |-  ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( Jt  Y )t  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )
113 ssexg 4341 . . . . . . . . 9  |-  ( ( Y  C_  RR  /\  RR  e.  _V )  ->  Y  e.  _V )
11413, 54, 113sylancl 644 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
115 restabs 17221 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) 
C_  Y  /\  Y  e.  _V )  ->  (
( Jt  Y )t  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
11652, 15, 114, 115syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( Jt  Y )t  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
117112, 116syl5eq 2479 . . . . . 6  |-  ( ph  ->  ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
118117oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( Nt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M )  =  ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) )
119118fveq1d 5722 . . . 4  |-  ( ph  ->  ( ( ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) )  =  ( ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) ) )
120110, 119eleqtrrd 2512 . . 3  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
12113, 39syl6ss 3352 . . . . . . 7  |-  ( ph  ->  Y  C_  CC )
122 resttopon 17217 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  Y  C_  CC )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
123101, 121, 122sylancr 645 . . . . . 6  |-  ( ph  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
124111, 123syl5eqel 2519 . . . . 5  |-  ( ph  ->  N  e.  (TopOn `  Y ) )
125 topontop 16983 . . . . 5  |-  ( N  e.  (TopOn `  Y
)  ->  N  e.  Top )
126124, 125syl 16 . . . 4  |-  ( ph  ->  N  e.  Top )
127 toponuni 16984 . . . . . 6  |-  ( N  e.  (TopOn `  Y
)  ->  Y  =  U. N )
128124, 127syl 16 . . . . 5  |-  ( ph  ->  Y  =  U. N
)
12915, 128sseqtrd 3376 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  U. N )
13015, 13sstrd 3350 . . . . . . . . 9  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  RR )
131 difssd 3467 . . . . . . . . 9  |-  ( ph  ->  ( RR  \  Y
)  C_  RR )
132130, 131unssd 3515 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) ) 
C_  RR )
133 ssun1 3502 . . . . . . . . 9  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) )
134133a1i 11 . . . . . . . 8  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )
13518ntrss 17111 . . . . . . . 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 ) ) ) )
1364, 132, 134, 135syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( int `  T
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
137136, 29sseldd 3341 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
138 f1of 5666 . . . . . . . 8  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
1395, 138syl 16 . . . . . . 7  |-  ( ph  ->  F : X --> Y )
140139, 23ffvelrnd 5863 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  Y )
141 elin 3522 . . . . . 6  |-  ( ( F `  C )  e.  ( ( ( int `  T ) `
 ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y )  <->  ( ( F `  C )  e.  ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  /\  ( F `  C )  e.  Y ) )
142137, 140, 141sylanbrc 646 . . . . 5  |-  ( ph  ->  ( F `  C
)  e.  ( ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y ) )
143 eqid 2435 . . . . . . . 8  |-  ( Tt  Y )  =  ( Tt  Y )
14418, 143restntr 17238 . . . . . . 7  |-  ( ( T  e.  Top  /\  Y  C_  RR  /\  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  Y )  ->  (
( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y ) )
1454, 13, 15, 144syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y ) )
146 restabs 17221 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  Y  C_  RR  /\  RR  e.  _V )  ->  (
( Jt  RR )t  Y )  =  ( Jt  Y ) )
14752, 13, 55, 146syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( Jt  RR )t  Y )  =  ( Jt  Y ) )
14849oveq1i 6083 . . . . . . . . 9  |-  ( Tt  Y )  =  ( ( Jt  RR )t  Y )
149147, 148, 1113eqtr4g 2492 . . . . . . . 8  |-  ( ph  ->  ( Tt  Y )  =  N )
150149fveq2d 5724 . . . . . . 7  |-  ( ph  ->  ( int `  ( Tt  Y ) )  =  ( int `  N
) )
151150fveq1d 5722 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
152145, 151eqtr3d 2469 . . . . 5  |-  ( ph  ->  ( ( ( int `  T ) `  (
( F " (
( C  -  R
) [,] ( C  +  R ) ) )  u.  ( RR 
\  Y ) ) )  i^i  Y )  =  ( ( int `  N ) `  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
153142, 152eleqtrd 2511 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
154128feq2d 5573 . . . . . 6  |-  ( ph  ->  ( `' F : Y
--> X  <->  `' F : U. N --> X ) )
15533, 154mpbid 202 . . . . 5  |-  ( ph  ->  `' F : U. N --> X )
156 resttopon 17217 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  CC )  /\  X  C_  CC )  ->  ( Jt  X )  e.  (TopOn `  X ) )
157101, 40, 156sylancr 645 . . . . . . 7  |-  ( ph  ->  ( Jt  X )  e.  (TopOn `  X ) )
15882, 157syl5eqel 2519 . . . . . 6  |-  ( ph  ->  M  e.  (TopOn `  X ) )
159 toponuni 16984 . . . . . 6  |-  ( M  e.  (TopOn `  X
)  ->  X  =  U. M )
160 feq3 5570 . . . . . 6  |-  ( X  =  U. M  -> 
( `' F : U. N --> X  <->  `' F : U. N --> U. M
) )
161158, 159, 1603syl 19 . . . . 5  |-  ( ph  ->  ( `' F : U. N --> X  <->  `' F : U. N --> U. M
) )
162155, 161mpbid 202 . . . 4  |-  ( ph  ->  `' F : U. N --> U. M )
163 eqid 2435 . . . . 5  |-  U. N  =  U. N
164 eqid 2435 . . . . 5  |-  U. M  =  U. M
165163, 164cnprest 17345 . . . 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.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) ) )
166126, 129, 153, 162, 165syl22anc 1185 . . 3  |-  ( ph  ->  ( `' F  e.  ( ( N  CnP  M ) `  ( F `
 C ) )  <-> 
( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) ) )
167120, 166mpbird 224 . 2  |-  ( ph  ->  `' F  e.  (
( N  CnP  M
) `  ( F `  C ) ) )
16830, 167jca 519 1  |-  ( ph  ->  ( ( F `  C )  e.  ( ( int `  T
) `  Y )  /\  `' F  e.  (
( N  CnP  M
) `  ( F `  C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   U.cuni 4007   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    <_ cle 9113    - cmin 9283   RR+crp 10604   (,)cioo 10908   [,]cicc 10911   ↾t crest 13640   TopOpenctopn 13641   topGenctg 13657  ℂfldccnfld 16695   Topctop 16950  TopOnctopon 16951   intcnt 17073    Cn ccn 17280    CnP ccnp 17281   Compccmp 17441   -cn->ccncf 18898    _D cdv 19742
This theorem is referenced by:  dvcnvre  19895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746
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