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Theorem dvcnvrelem2 19365
Description: Lemma for dvcnvre 19366. (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 18270 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
31, 2eqeltri 2353 . . . . 5  |-  T  e. 
Top
43a1i 10 . . . 4  |-  ( ph  ->  T  e.  Top )
5 dvcnvre.1 . . . . . 6  |-  ( ph  ->  F : X -1-1-onto-> Y )
6 f1ofo 5479 . . . . . 6  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
7 forn 5454 . . . . . 6  |-  ( F : X -onto-> Y  ->  ran  F  =  Y )
85, 6, 73syl 18 . . . . 5  |-  ( ph  ->  ran  F  =  Y )
9 dvcnvre.f . . . . . 6  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
10 cncff 18397 . . . . . 6  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
11 frn 5395 . . . . . 6  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
129, 10, 113syl 18 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
138, 12eqsstr3d 3213 . . . 4  |-  ( ph  ->  Y  C_  RR )
14 imassrn 5025 . . . . 5  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ran  F
1514, 8syl5sseq 3226 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  Y )
16 uniretop 18271 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
171unieqi 3837 . . . . . 6  |-  U. T  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2306 . . . . 5  |-  RR  =  U. T
1918ntrss 16792 . . . 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 1182 . . 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 19364 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
271fveq2i 5528 . . . . 5  |-  ( int `  T )  =  ( int `  ( topGen ` 
ran  (,) ) )
2827fveq1i 5526 . . . 4  |-  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )
2926, 28syl6eleqr 2374 . . 3  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
3020, 29sseldd 3181 . 2  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 Y ) )
31 f1ocnv 5485 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
32 f1of 5472 . . . . . . 7  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
335, 31, 323syl 18 . . . . . 6  |-  ( ph  ->  `' F : Y --> X )
34 ffun 5391 . . . . . 6  |-  ( `' F : Y --> X  ->  Fun  `' F )
35 funcnvres 5321 . . . . . 6  |-  ( Fun  `' F  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
3633, 34, 353syl 18 . . . . 5  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( `' F  |`  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
37 dvbsss 19252 . . . . . . . . . . . 12  |-  dom  ( RR  _D  F )  C_  RR
3837a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
3921, 38eqsstr3d 3213 . . . . . . . . . 10  |-  ( ph  ->  X  C_  RR )
40 ax-resscn 8794 . . . . . . . . . 10  |-  RR  C_  CC
4139, 40syl6ss 3191 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
42 cncfss 18403 . . . . . . . . 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 ) )
4325, 41, 42syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) -cn-> ( ( C  -  R ) [,] ( C  +  R ) ) ) 
C_  ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) -cn-> X ) )
44 f1of1 5471 . . . . . . . . . . 11  |-  ( F : X -1-1-onto-> Y  ->  F : X -1-1-> Y )
455, 44syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : X -1-1-> Y
)
46 f1ores 5487 . . . . . . . . . 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
) ) ) )
4745, 25, 46syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) : ( ( C  -  R ) [,] ( C  +  R ) ) -1-1-onto-> ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )
48 dvcnvre.j . . . . . . . . . . . . . . 15  |-  J  =  ( TopOpen ` fld )
4948tgioo2 18309 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( Jt  RR )
501, 49eqtri 2303 . . . . . . . . . . . . 13  |-  T  =  ( Jt  RR )
5150oveq1i 5868 . . . . . . . . . . . 12  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( ( Jt  RR )t  ( ( C  -  R ) [,] ( C  +  R
) ) )
5248cnfldtop 18293 . . . . . . . . . . . . . 14  |-  J  e. 
Top
5352a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
5425, 39sstrd 3189 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
55 reex 8828 . . . . . . . . . . . . . 14  |-  RR  e.  _V
5655a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  e.  _V )
57 restabs 16896 . . . . . . . . . . . . 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
) ) ) )
5853, 54, 56, 57syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( Jt  RR )t  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) ) )
5951, 58syl5eq 2327 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) ) )
6039, 23sseldd 3181 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
6124rpred 10390 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
6260, 61resubcld 9211 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR )
6360, 61readdcld 8862 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR )
64 eqid 2283 . . . . . . . . . . . . 13  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )
651, 64icccmp 18330 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R ) ) )  e.  Comp )
6662, 63, 65syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
6759, 66eqeltrrd 2358 . . . . . . . . . 10  |-  ( ph  ->  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
68 f1of 5472 . . . . . . . . . . . 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 ) ) ) )
6947, 68syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) : ( ( C  -  R ) [,] ( C  +  R ) ) --> ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
7012, 40syl6ss 3191 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  F  C_  CC )
7114, 70syl5ss 3190 . . . . . . . . . . . 12  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  CC )
72 rescncf 18401 . . . . . . . . . . . . 13  |-  ( ( ( C  -  R
) [,] ( C  +  R ) ) 
C_  X  ->  ( F  e.  ( X -cn->
RR )  ->  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  e.  ( ( ( C  -  R ) [,] ( C  +  R
) ) -cn-> RR ) ) )
7325, 9, 72sylc 56 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> RR ) )
74 cncffvrn 18402 . . . . . . . . . . . 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 ) ) ) ) )
7571, 73, 74syl2anc 642 . . . . . . . . . . 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 ) ) ) ) )
7669, 75mpbird 223 . . . . . . . . . 10  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) ) )
77 eqid 2283 . . . . . . . . . . 11  |-  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )
7848, 77cncfcnvcn 18424 . . . . . . . . . 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
) ) ) ) )
7967, 76, 78syl2anc 642 . . . . . . . . 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
) ) ) ) )
8047, 79mpbid 201 . . . . . . . 8  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> ( ( C  -  R ) [,] ( C  +  R
) ) ) )
8143, 80sseldd 3181 . . . . . . 7  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> X ) )
82 eqid 2283 . . . . . . . . 9  |-  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
83 dvcnvre.m . . . . . . . . 9  |-  M  =  ( Jt  X )
8448, 82, 83cncfcn 18413 . . . . . . . 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 ) )
8571, 41, 84syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) -cn-> X )  =  ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  Cn  M ) )
8681, 85eleqtrd 2359 . . . . . 6  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  Cn  M ) )
8760, 24ltsubrpd 10418 . . . . . . . . . 10  |-  ( ph  ->  ( C  -  R
)  <  C )
8862, 60, 87ltled 8967 . . . . . . . . 9  |-  ( ph  ->  ( C  -  R
)  <_  C )
8960, 24ltaddrpd 10419 . . . . . . . . . 10  |-  ( ph  ->  C  <  ( C  +  R ) )
9060, 63, 89ltled 8967 . . . . . . . . 9  |-  ( ph  ->  C  <_  ( C  +  R ) )
91 elicc2 10715 . . . . . . . . . 10  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  <->  ( C  e.  RR  /\  ( C  -  R )  <_  C  /\  C  <_  ( C  +  R )
) ) )
9262, 63, 91syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  <-> 
( C  e.  RR  /\  ( C  -  R
)  <_  C  /\  C  <_  ( C  +  R ) ) ) )
9360, 88, 90, 92mpbir3and 1135 . . . . . . . 8  |-  ( ph  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
94 ffun 5391 . . . . . . . . . 10  |-  ( F : X --> RR  ->  Fun 
F )
959, 10, 943syl 18 . . . . . . . . 9  |-  ( ph  ->  Fun  F )
96 fdm 5393 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  dom 
F  =  X )
979, 10, 963syl 18 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  X )
9825, 97sseqtr4d 3215 . . . . . . . . 9  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  dom  F )
99 funfvima2 5754 . . . . . . . . 9  |-  ( ( Fun  F  /\  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  C )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
10095, 98, 99syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  C )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
10193, 100mpd 14 . . . . . . 7  |-  ( ph  ->  ( F `  C
)  e.  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )
10248cnfldtopon 18292 . . . . . . . . 9  |-  J  e.  (TopOn `  CC )
103 resttopon 16892 . . . . . . . . 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 ) ) ) ) )
104102, 71, 103sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  e.  (TopOn `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) ) )
105 toponuni 16665 . . . . . . . 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
) ) ) ) )
106104, 105syl 15 . . . . . . 7  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
107101, 106eleqtrd 2359 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
108 eqid 2283 . . . . . . 7  |-  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
109108cncnpi 17007 . . . . . 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 ) ) )
11086, 107, 109syl2anc 642 . . . . 5  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) `  ( F `  C ) ) )
11136, 110eqeltrrd 2358 . . . 4  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
112 dvcnvre.n . . . . . . . 8  |-  N  =  ( Jt  Y )
113112oveq1i 5868 . . . . . . 7  |-  ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( Jt  Y )t  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )
114 ssexg 4160 . . . . . . . . 9  |-  ( ( Y  C_  RR  /\  RR  e.  _V )  ->  Y  e.  _V )
11513, 55, 114sylancl 643 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
116 restabs 16896 . . . . . . . 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 )
) ) ) )
11753, 15, 115, 116syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( Jt  Y )t  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
118113, 117syl5eq 2327 . . . . . 6  |-  ( ph  ->  ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
119118oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( Nt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M )  =  ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) )
120119fveq1d 5527 . . . 4  |-  ( ph  ->  ( ( ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) )  =  ( ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) ) )
121111, 120eleqtrrd 2360 . . 3  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
12213, 40syl6ss 3191 . . . . . . 7  |-  ( ph  ->  Y  C_  CC )
123 resttopon 16892 . . . . . . 7  |-  ( ( J  e.  (TopOn `  CC )  /\  Y  C_  CC )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
124102, 122, 123sylancr 644 . . . . . 6  |-  ( ph  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
125112, 124syl5eqel 2367 . . . . 5  |-  ( ph  ->  N  e.  (TopOn `  Y ) )
126 topontop 16664 . . . . 5  |-  ( N  e.  (TopOn `  Y
)  ->  N  e.  Top )
127125, 126syl 15 . . . 4  |-  ( ph  ->  N  e.  Top )
128 toponuni 16665 . . . . . 6  |-  ( N  e.  (TopOn `  Y
)  ->  Y  =  U. N )
129125, 128syl 15 . . . . 5  |-  ( ph  ->  Y  =  U. N
)
13015, 129sseqtrd 3214 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  U. N )
13115, 13sstrd 3189 . . . . . . . . 9  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  RR )
132 difss 3303 . . . . . . . . . 10  |-  ( RR 
\  Y )  C_  RR
133132a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( RR  \  Y
)  C_  RR )
134131, 133unssd 3351 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) ) 
C_  RR )
135 ssun1 3338 . . . . . . . . 9  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) )
136135a1i 10 . . . . . . . 8  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )
13718ntrss 16792 . . . . . . . 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 ) ) ) )
1384, 134, 136, 137syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( int `  T
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
139138, 29sseldd 3181 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
140 f1of 5472 . . . . . . . 8  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
1415, 140syl 15 . . . . . . 7  |-  ( ph  ->  F : X --> Y )
142 ffvelrn 5663 . . . . . . 7  |-  ( ( F : X --> Y  /\  C  e.  X )  ->  ( F `  C
)  e.  Y )
143141, 23, 142syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  Y )
144 elin 3358 . . . . . 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 ) )
145139, 143, 144sylanbrc 645 . . . . 5  |-  ( ph  ->  ( F `  C
)  e.  ( ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y ) )
146 eqid 2283 . . . . . . . 8  |-  ( Tt  Y )  =  ( Tt  Y )
14718, 146restntr 16912 . . . . . . 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 ) )
1484, 13, 15, 147syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( ( int `  T
) `  ( ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) )  i^i 
Y ) )
149 restabs 16896 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  Y  C_  RR  /\  RR  e.  _V )  ->  (
( Jt  RR )t  Y )  =  ( Jt  Y ) )
15053, 13, 56, 149syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( Jt  RR )t  Y )  =  ( Jt  Y ) )
15150oveq1i 5868 . . . . . . . . 9  |-  ( Tt  Y )  =  ( ( Jt  RR )t  Y )
152150, 151, 1123eqtr4g 2340 . . . . . . . 8  |-  ( ph  ->  ( Tt  Y )  =  N )
153152fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( int `  ( Tt  Y ) )  =  ( int `  N
) )
154153fveq1d 5527 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
155148, 154eqtr3d 2317 . . . . 5  |-  ( ph  ->  ( ( ( int `  T ) `  (
( F " (
( C  -  R
) [,] ( C  +  R ) ) )  u.  ( RR 
\  Y ) ) )  i^i  Y )  =  ( ( int `  N ) `  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
156145, 155eleqtrd 2359 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
157129feq2d 5380 . . . . . 6  |-  ( ph  ->  ( `' F : Y
--> X  <->  `' F : U. N --> X ) )
15833, 157mpbid 201 . . . . 5  |-  ( ph  ->  `' F : U. N --> X )
159 resttopon 16892 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  CC )  /\  X  C_  CC )  ->  ( Jt  X )  e.  (TopOn `  X ) )
160102, 41, 159sylancr 644 . . . . . . 7  |-  ( ph  ->  ( Jt  X )  e.  (TopOn `  X ) )
16183, 160syl5eqel 2367 . . . . . 6  |-  ( ph  ->  M  e.  (TopOn `  X ) )
162 toponuni 16665 . . . . . 6  |-  ( M  e.  (TopOn `  X
)  ->  X  =  U. M )
163 feq3 5377 . . . . . 6  |-  ( X  =  U. M  -> 
( `' F : U. N --> X  <->  `' F : U. N --> U. M
) )
164161, 162, 1633syl 18 . . . . 5  |-  ( ph  ->  ( `' F : U. N --> X  <->  `' F : U. N --> U. M
) )
165158, 164mpbid 201 . . . 4  |-  ( ph  ->  `' F : U. N --> U. M )
166 eqid 2283 . . . . 5  |-  U. N  =  U. N
167 eqid 2283 . . . . 5  |-  U. M  =  U. M
168166, 167cnprest 17017 . . . 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 )
) ) )
169127, 130, 156, 165, 168syl22anc 1183 . . 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 )
) ) )
170121, 169mpbird 223 . 2  |-  ( ph  ->  `' F  e.  (
( N  CnP  M
) `  ( F `  C ) ) )
17130, 170jca 518 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   U.cuni 3827   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    <_ cle 8868    - cmin 9037   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   ↾t crest 13325   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632   intcnt 16754    Cn ccn 16954    CnP ccnp 16955   Compccmp 17113   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvcnvre  19366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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