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Theorem dvcnvrelem2 19855
Description: Lemma for dvcnvre 19856. (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 18748 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
31, 2eqeltri 2474 . . . . 5  |-  T  e. 
Top
43a1i 11 . . . 4  |-  ( ph  ->  T  e.  Top )
5 dvcnvre.1 . . . . . 6  |-  ( ph  ->  F : X -1-1-onto-> Y )
6 f1ofo 5640 . . . . . 6  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
7 forn 5615 . . . . . 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 18876 . . . . . 6  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
11 frn 5556 . . . . . 6  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
129, 10, 113syl 19 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
138, 12eqsstr3d 3343 . . . 4  |-  ( ph  ->  Y  C_  RR )
14 imassrn 5175 . . . . 5  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ran  F
1514, 8syl5sseq 3356 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  Y )
16 uniretop 18749 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
171unieqi 3985 . . . . . 6  |-  U. T  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2427 . . . . 5  |-  RR  =  U. T
1918ntrss 17074 . . . 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 19854 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
271fveq2i 5690 . . . . 5  |-  ( int `  T )  =  ( int `  ( topGen ` 
ran  (,) ) )
2827fveq1i 5688 . . . 4  |-  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )
2926, 28syl6eleqr 2495 . . 3  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
3020, 29sseldd 3309 . 2  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 Y ) )
31 f1ocnv 5646 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
32 f1of 5633 . . . . . . 7  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
335, 31, 323syl 19 . . . . . 6  |-  ( ph  ->  `' F : Y --> X )
34 ffun 5552 . . . . . 6  |-  ( `' F : Y --> X  ->  Fun  `' F )
35 funcnvres 5481 . . . . . 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 19742 . . . . . . . . . . 11  |-  dom  ( RR  _D  F )  C_  RR
3821, 37syl6eqssr 3359 . . . . . . . . . 10  |-  ( ph  ->  X  C_  RR )
39 ax-resscn 9003 . . . . . . . . . 10  |-  RR  C_  CC
4038, 39syl6ss 3320 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
41 cncfss 18882 . . . . . . . . 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 5632 . . . . . . . . . . 11  |-  ( F : X -1-1-onto-> Y  ->  F : X -1-1-> Y )
445, 43syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : X -1-1-> Y
)
45 f1ores 5648 . . . . . . . . . 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 18787 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( Jt  RR )
491, 48eqtri 2424 . . . . . . . . . . . . 13  |-  T  =  ( Jt  RR )
5049oveq1i 6050 . . . . . . . . . . . 12  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( ( Jt  RR )t  ( ( C  -  R ) [,] ( C  +  R
) ) )
5147cnfldtop 18771 . . . . . . . . . . . . . 14  |-  J  e. 
Top
5251a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
5325, 38sstrd 3318 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
54 reex 9037 . . . . . . . . . . . . . 14  |-  RR  e.  _V
5554a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  e.  _V )
56 restabs 17183 . . . . . . . . . . . . 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 2448 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) ) )
5938, 23sseldd 3309 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
6024rpred 10604 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
6159, 60resubcld 9421 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR )
6259, 60readdcld 9071 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR )
63 eqid 2404 . . . . . . . . . . . . 13  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )
641, 63icccmp 18809 . . . . . . . . . . . 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 2479 . . . . . . . . . 10  |-  ( ph  ->  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
67 f1of 5633 . . . . . . . . . . . 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 3320 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  F  C_  CC )
7014, 69syl5ss 3319 . . . . . . . . . . . 12  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  CC )
71 rescncf 18880 . . . . . . . . . . . . 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 18881 . . . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )
7747, 76cncfcnvcn 18904 . . . . . . . . . 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 3309 . . . . . . 7  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> X ) )
81 eqid 2404 . . . . . . . . 9  |-  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
82 dvcnvre.m . . . . . . . . 9  |-  M  =  ( Jt  X )
8347, 81, 82cncfcn 18892 . . . . . . . 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 2480 . . . . . 6  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  Cn  M ) )
8659, 24ltsubrpd 10632 . . . . . . . . . 10  |-  ( ph  ->  ( C  -  R
)  <  C )
8761, 59, 86ltled 9177 . . . . . . . . 9  |-  ( ph  ->  ( C  -  R
)  <_  C )
8859, 24ltaddrpd 10633 . . . . . . . . . 10  |-  ( ph  ->  C  <  ( C  +  R ) )
8959, 62, 88ltled 9177 . . . . . . . . 9  |-  ( ph  ->  C  <_  ( C  +  R ) )
90 elicc2 10931 . . . . . . . . . 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 5552 . . . . . . . . . 10  |-  ( F : X --> RR  ->  Fun 
F )
949, 10, 933syl 19 . . . . . . . . 9  |-  ( ph  ->  Fun  F )
95 fdm 5554 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  dom 
F  =  X )
969, 10, 953syl 19 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  X )
9725, 96sseqtr4d 3345 . . . . . . . . 9  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  dom  F )
98 funfvima2 5933 . . . . . . . . 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 18770 . . . . . . . . 9  |-  J  e.  (TopOn `  CC )
102 resttopon 17179 . . . . . . . . 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 16947 . . . . . . . 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 2480 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
107 eqid 2404 . . . . . . 7  |-  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
108107cncnpi 17296 . . . . . 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 2479 . . . 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 6050 . . . . . . 7  |-  ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( Jt  Y )t  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )
113 ssexg 4309 . . . . . . . . 9  |-  ( ( Y  C_  RR  /\  RR  e.  _V )  ->  Y  e.  _V )
11413, 54, 113sylancl 644 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
115 restabs 17183 . . . . . . . 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 2448 . . . . . 6  |-  ( ph  ->  ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
118117oveq1d 6055 . . . . 5  |-  ( ph  ->  ( ( Nt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M )  =  ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) )
119118fveq1d 5689 . . . 4  |-  ( ph  ->  ( ( ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) )  =  ( ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) ) )
120110, 119eleqtrrd 2481 . . 3  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
12113, 39syl6ss 3320 . . . . . . 7  |-  ( ph  ->  Y  C_  CC )
122 resttopon 17179 . . . . . . 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 2488 . . . . 5  |-  ( ph  ->  N  e.  (TopOn `  Y ) )
125 topontop 16946 . . . . 5  |-  ( N  e.  (TopOn `  Y
)  ->  N  e.  Top )
126124, 125syl 16 . . . 4  |-  ( ph  ->  N  e.  Top )
127 toponuni 16947 . . . . . 6  |-  ( N  e.  (TopOn `  Y
)  ->  Y  =  U. N )
128124, 127syl 16 . . . . 5  |-  ( ph  ->  Y  =  U. N
)
12915, 128sseqtrd 3344 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  U. N )
13015, 13sstrd 3318 . . . . . . . . 9  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  RR )
131 difssd 3435 . . . . . . . . 9  |-  ( ph  ->  ( RR  \  Y
)  C_  RR )
132130, 131unssd 3483 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) ) 
C_  RR )
133 ssun1 3470 . . . . . . . . 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 17074 . . . . . . . 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 3309 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
138 f1of 5633 . . . . . . . 8  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
1395, 138syl 16 . . . . . . 7  |-  ( ph  ->  F : X --> Y )
140139, 23ffvelrnd 5830 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  Y )
141 elin 3490 . . . . . 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 2404 . . . . . . . 8  |-  ( Tt  Y )  =  ( Tt  Y )
14418, 143restntr 17200 . . . . . . 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 17183 . . . . . . . . . 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 6050 . . . . . . . . 9  |-  ( Tt  Y )  =  ( ( Jt  RR )t  Y )
149147, 148, 1113eqtr4g 2461 . . . . . . . 8  |-  ( ph  ->  ( Tt  Y )  =  N )
150149fveq2d 5691 . . . . . . 7  |-  ( ph  ->  ( int `  ( Tt  Y ) )  =  ( int `  N
) )
151150fveq1d 5689 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
152145, 151eqtr3d 2438 . . . . 5  |-  ( ph  ->  ( ( ( int `  T ) `  (
( F " (
( C  -  R
) [,] ( C  +  R ) ) )  u.  ( RR 
\  Y ) ) )  i^i  Y )  =  ( ( int `  N ) `  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
153142, 152eleqtrd 2480 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
154128feq2d 5540 . . . . . 6  |-  ( ph  ->  ( `' F : Y
--> X  <->  `' F : U. N --> X ) )
15533, 154mpbid 202 . . . . 5  |-  ( ph  ->  `' F : U. N --> X )
156 resttopon 17179 . . . . . . . 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 2488 . . . . . 6  |-  ( ph  ->  M  e.  (TopOn `  X ) )
159 toponuni 16947 . . . . . 6  |-  ( M  e.  (TopOn `  X
)  ->  X  =  U. M )
160 feq3 5537 . . . . . 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 2404 . . . . 5  |-  U. N  =  U. N
164 eqid 2404 . . . . 5  |-  U. M  =  U. M
165163, 164cnprest 17307 . . . 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 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   U.cuni 3975   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    <_ cle 9077    - cmin 9247   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914   intcnt 17036    Cn ccn 17242    CnP ccnp 17243   Compccmp 17403   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvcnvre  19856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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