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Theorem dvcnvrelem2 19381
Description: Lemma for dvcnvre 19382. (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 18286 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
31, 2eqeltri 2366 . . . . 5  |-  T  e. 
Top
43a1i 10 . . . 4  |-  ( ph  ->  T  e.  Top )
5 dvcnvre.1 . . . . . 6  |-  ( ph  ->  F : X -1-1-onto-> Y )
6 f1ofo 5495 . . . . . 6  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
7 forn 5470 . . . . . 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 18413 . . . . . 6  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
11 frn 5411 . . . . . 6  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
129, 10, 113syl 18 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
138, 12eqsstr3d 3226 . . . 4  |-  ( ph  ->  Y  C_  RR )
14 imassrn 5041 . . . . 5  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  C_  ran  F
1514, 8syl5sseq 3239 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  Y )
16 uniretop 18287 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
171unieqi 3853 . . . . . 6  |-  U. T  =  U. ( topGen `  ran  (,) )
1816, 17eqtr4i 2319 . . . . 5  |-  RR  =  U. T
1918ntrss 16808 . . . 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 19380 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
271fveq2i 5544 . . . . 5  |-  ( int `  T )  =  ( int `  ( topGen ` 
ran  (,) ) )
2827fveq1i 5542 . . . 4  |-  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )
2926, 28syl6eleqr 2387 . . 3  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
3020, 29sseldd 3194 . 2  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 Y ) )
31 f1ocnv 5501 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
32 f1of 5488 . . . . . . 7  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
335, 31, 323syl 18 . . . . . 6  |-  ( ph  ->  `' F : Y --> X )
34 ffun 5407 . . . . . 6  |-  ( `' F : Y --> X  ->  Fun  `' F )
35 funcnvres 5337 . . . . . 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 19268 . . . . . . . . . . . 12  |-  dom  ( RR  _D  F )  C_  RR
3837a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
3921, 38eqsstr3d 3226 . . . . . . . . . 10  |-  ( ph  ->  X  C_  RR )
40 ax-resscn 8810 . . . . . . . . . 10  |-  RR  C_  CC
4139, 40syl6ss 3204 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
42 cncfss 18419 . . . . . . . . 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 5487 . . . . . . . . . . 11  |-  ( F : X -1-1-onto-> Y  ->  F : X -1-1-> Y )
455, 44syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : X -1-1-> Y
)
46 f1ores 5503 . . . . . . . . . 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 18325 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( Jt  RR )
501, 49eqtri 2316 . . . . . . . . . . . . 13  |-  T  =  ( Jt  RR )
5150oveq1i 5884 . . . . . . . . . . . 12  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( ( Jt  RR )t  ( ( C  -  R ) [,] ( C  +  R
) ) )
5248cnfldtop 18309 . . . . . . . . . . . . . 14  |-  J  e. 
Top
5352a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
5425, 39sstrd 3202 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
55 reex 8844 . . . . . . . . . . . . . 14  |-  RR  e.  _V
5655a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  e.  _V )
57 restabs 16912 . . . . . . . . . . . . 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 2340 . . . . . . . . . . 11  |-  ( ph  ->  ( Tt  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) ) )
6039, 23sseldd 3194 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
6124rpred 10406 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
6260, 61resubcld 9227 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR )
6360, 61readdcld 8878 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR )
64 eqid 2296 . . . . . . . . . . . . 13  |-  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Tt  ( ( C  -  R
) [,] ( C  +  R ) ) )
651, 64icccmp 18346 . . . . . . . . . . . 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 2371 . . . . . . . . . 10  |-  ( ph  ->  ( Jt  ( ( C  -  R ) [,] ( C  +  R
) ) )  e. 
Comp )
68 f1of 5488 . . . . . . . . . . . 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 3204 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  F  C_  CC )
7114, 70syl5ss 3203 . . . . . . . . . . . 12  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  CC )
72 rescncf 18417 . . . . . . . . . . . . 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 18418 . . . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )  =  ( Jt  ( ( C  -  R
) [,] ( C  +  R ) ) )
7848, 77cncfcnvcn 18440 . . . . . . . . . 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 3194 . . . . . . 7  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) -cn-> X ) )
82 eqid 2296 . . . . . . . . 9  |-  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
83 dvcnvre.m . . . . . . . . 9  |-  M  =  ( Jt  X )
8448, 82, 83cncfcn 18429 . . . . . . . 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 2372 . . . . . 6  |-  ( ph  ->  `' ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  e.  ( ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  Cn  M ) )
8760, 24ltsubrpd 10434 . . . . . . . . . 10  |-  ( ph  ->  ( C  -  R
)  <  C )
8862, 60, 87ltled 8983 . . . . . . . . 9  |-  ( ph  ->  ( C  -  R
)  <_  C )
8960, 24ltaddrpd 10435 . . . . . . . . . 10  |-  ( ph  ->  C  <  ( C  +  R ) )
9060, 63, 89ltled 8983 . . . . . . . . 9  |-  ( ph  ->  C  <_  ( C  +  R ) )
91 elicc2 10731 . . . . . . . . . 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 5407 . . . . . . . . . 10  |-  ( F : X --> RR  ->  Fun 
F )
959, 10, 943syl 18 . . . . . . . . 9  |-  ( ph  ->  Fun  F )
96 fdm 5409 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  dom 
F  =  X )
979, 10, 963syl 18 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  X )
9825, 97sseqtr4d 3228 . . . . . . . . 9  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  dom  F )
99 funfvima2 5770 . . . . . . . . 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 18308 . . . . . . . . 9  |-  J  e.  (TopOn `  CC )
103 resttopon 16908 . . . . . . . . 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 16681 . . . . . . . 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 2372 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
108 eqid 2296 . . . . . . 7  |-  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  U. ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )
109108cncnpi 17023 . . . . . 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 2371 . . . 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 5884 . . . . . . 7  |-  ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( Jt  Y )t  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )
114 ssexg 4176 . . . . . . . . 9  |-  ( ( Y  C_  RR  /\  RR  e.  _V )  ->  Y  e.  _V )
11513, 55, 114sylancl 643 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
116 restabs 16912 . . . . . . . 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 2340 . . . . . 6  |-  ( ph  ->  ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( Jt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
119118oveq1d 5889 . . . . 5  |-  ( ph  ->  ( ( Nt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M )  =  ( ( Jt  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) )  CnP  M ) )
120119fveq1d 5543 . . . 4  |-  ( ph  ->  ( ( ( Nt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) )  =  ( ( ( Jt  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  CnP  M ) `
 ( F `  C ) ) )
121111, 120eleqtrrd 2373 . . 3  |-  ( ph  ->  ( `' F  |`  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  e.  ( ( ( Nt  ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) )  CnP 
M ) `  ( F `  C )
) )
12213, 40syl6ss 3204 . . . . . . 7  |-  ( ph  ->  Y  C_  CC )
123 resttopon 16908 . . . . . . 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 2380 . . . . 5  |-  ( ph  ->  N  e.  (TopOn `  Y ) )
126 topontop 16680 . . . . 5  |-  ( N  e.  (TopOn `  Y
)  ->  N  e.  Top )
127125, 126syl 15 . . . 4  |-  ( ph  ->  N  e.  Top )
128 toponuni 16681 . . . . . 6  |-  ( N  e.  (TopOn `  Y
)  ->  Y  =  U. N )
129125, 128syl 15 . . . . 5  |-  ( ph  ->  Y  =  U. N
)
13015, 129sseqtrd 3227 . . . 4  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  U. N )
13115, 13sstrd 3202 . . . . . . . . 9  |-  ( ph  ->  ( F " (
( C  -  R
) [,] ( C  +  R ) ) )  C_  RR )
132 difss 3316 . . . . . . . . . 10  |-  ( RR 
\  Y )  C_  RR
133132a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( RR  \  Y
)  C_  RR )
134131, 133unssd 3364 . . . . . . . 8  |-  ( ph  ->  ( ( F "
( ( C  -  R ) [,] ( C  +  R )
) )  u.  ( RR  \  Y ) ) 
C_  RR )
135 ssun1 3351 . . . . . . . . 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 16808 . . . . . . . 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 3194 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  T ) `
 ( ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  u.  ( RR  \  Y
) ) ) )
140 f1of 5488 . . . . . . . 8  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
1415, 140syl 15 . . . . . . 7  |-  ( ph  ->  F : X --> Y )
142 ffvelrn 5679 . . . . . . 7  |-  ( ( F : X --> Y  /\  C  e.  X )  ->  ( F `  C
)  e.  Y )
143141, 23, 142syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F `  C
)  e.  Y )
144 elin 3371 . . . . . 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 2296 . . . . . . . 8  |-  ( Tt  Y )  =  ( Tt  Y )
14718, 146restntr 16928 . . . . . . 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 16912 . . . . . . . . . 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 5884 . . . . . . . . 9  |-  ( Tt  Y )  =  ( ( Jt  RR )t  Y )
152150, 151, 1123eqtr4g 2353 . . . . . . . 8  |-  ( ph  ->  ( Tt  Y )  =  N )
153152fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( int `  ( Tt  Y ) )  =  ( int `  N
) )
154153fveq1d 5543 . . . . . 6  |-  ( ph  ->  ( ( int `  ( Tt  Y ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
155148, 154eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( ( ( int `  T ) `  (
( F " (
( C  -  R
) [,] ( C  +  R ) ) )  u.  ( RR 
\  Y ) ) )  i^i  Y )  =  ( ( int `  N ) `  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
156145, 155eleqtrd 2372 . . . 4  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  N ) `
 ( F "
( ( C  -  R ) [,] ( C  +  R )
) ) ) )
157129feq2d 5396 . . . . . 6  |-  ( ph  ->  ( `' F : Y
--> X  <->  `' F : U. N --> X ) )
15833, 157mpbid 201 . . . . 5  |-  ( ph  ->  `' F : U. N --> X )
159 resttopon 16908 . . . . . . . 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 2380 . . . . . 6  |-  ( ph  ->  M  e.  (TopOn `  X ) )
162 toponuni 16681 . . . . . 6  |-  ( M  e.  (TopOn `  X
)  ->  X  =  U. M )
163 feq3 5393 . . . . . 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 2296 . . . . 5  |-  U. N  =  U. N
167 eqid 2296 . . . . 5  |-  U. M  =  U. M
168166, 167cnprest 17033 . . . 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   U.cuni 3843   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    <_ cle 8884    - cmin 9053   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648   intcnt 16770    Cn ccn 16970    CnP ccnp 16971   Compccmp 17129   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvcnvre  19382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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