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Theorem i1fd 22214
Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fd.1  |-  ( ph  ->  F : RR --> RR )
i1fd.2  |-  ( ph  ->  ran  F  e.  Fin )
i1fd.3  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
i1fd.4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
Assertion
Ref Expression
i1fd  |-  ( ph  ->  F  e.  dom  S.1 )
Distinct variable groups:    x, F    ph, x

Proof of Theorem i1fd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 i1fd.1 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
21ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  F : RR --> RR )
3 ffun 5739 . . . . . . . 8  |-  ( F : RR --> RR  ->  Fun 
F )
4 funcnvcnv 5652 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  `' `' F )
5 imadif 5669 . . . . . . . 8  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
62, 3, 4, 54syl 21 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
7 ioof 11647 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 frn 5743 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  ran  (,)  C_ 
~P RR
109sseli 3495 . . . . . . . . . . 11  |-  ( x  e.  ran  (,)  ->  x  e.  ~P RR )
1110elpwid 4025 . . . . . . . . . 10  |-  ( x  e.  ran  (,)  ->  x 
C_  RR )
1211ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  x  C_  RR )
13 dfss4 3739 . . . . . . . . 9  |-  ( x 
C_  RR  <->  ( RR  \ 
( RR  \  x
) )  =  x )
1412, 13sylib 196 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( RR  \  ( RR  \  x ) )  =  x )
1514imaeq2d 5347 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( `' F " x ) )
166, 15eqtr3d 2500 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  =  ( `' F " x ) )
17 fimacnv 6020 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ( `' F " RR )  =  RR )
182, 17syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  =  RR )
19 rembl 22077 . . . . . . . 8  |-  RR  e.  dom  vol
2018, 19syl6eqel 2553 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  e.  dom  vol )
211adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  F : RR --> RR )
22 inpreima 6015 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( `' F " ( y  i^i 
ran  F ) )  =  ( ( `' F " y )  i^i  ( `' F " ran  F ) ) )
23 iunid 4387 . . . . . . . . . . . . . . . 16  |-  U_ x  e.  ( y  i^i  ran  F ) { x }  =  ( y  i^i 
ran  F )
2423imaeq2i 5345 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  ( `' F " ( y  i^i  ran  F )
)
25 imaiun 6158 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
2624, 25eqtr3i 2488 . . . . . . . . . . . . . 14  |-  ( `' F " ( y  i^i  ran  F )
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
27 cnvimass 5367 . . . . . . . . . . . . . . . 16  |-  ( `' F " y ) 
C_  dom  F
28 cnvimarndm 5368 . . . . . . . . . . . . . . . 16  |-  ( `' F " ran  F
)  =  dom  F
2927, 28sseqtr4i 3532 . . . . . . . . . . . . . . 15  |-  ( `' F " y ) 
C_  ( `' F " ran  F )
30 df-ss 3485 . . . . . . . . . . . . . . 15  |-  ( ( `' F " y ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y ) )
3129, 30mpbi 208 . . . . . . . . . . . . . 14  |-  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y )
3222, 26, 313eqtr3g 2521 . . . . . . . . . . . . 13  |-  ( Fun 
F  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
3321, 3, 323syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
34 i1fd.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  e.  Fin )
3534adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  0  e.  y )  ->  ran  F  e.  Fin )
36 inss2 3715 . . . . . . . . . . . . . 14  |-  ( y  i^i  ran  F )  C_ 
ran  F
37 ssfi 7759 . . . . . . . . . . . . . 14  |-  ( ( ran  F  e.  Fin  /\  ( y  i^i  ran  F )  C_  ran  F )  ->  ( y  i^i 
ran  F )  e. 
Fin )
3835, 36, 37sylancl 662 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F )  e.  Fin )
39 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ph )
40 inss1 3714 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  i^i  ran  F )  C_  y
4140sseli 3495 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( y  i^i 
ran  F )  -> 
0  e.  y )
4241con3i 135 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  0  e.  y  ->  -.  0  e.  (
y  i^i  ran  F ) )
4342adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  0  e.  y )  ->  -.  0  e.  ( y  i^i  ran  F ) )
44 disjsn 4092 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( y  i^i  ran  F ) )
4543, 44sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
( y  i^i  ran  F )  i^i  { 0 } )  =  (/) )
46 reldisj 3873 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  i^i  ran  F
)  C_  ran  F  -> 
( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  ( y  i^i 
ran  F )  C_  ( ran  F  \  {
0 } ) ) )
4736, 46ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4845, 47sylib 196 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4948sselda 3499 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  x  e.  ( ran  F  \  {
0 } ) )
50 i1fd.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
5139, 49, 50syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  e.  dom  vol )
5251ralrimiva 2871 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
53 finiunmbl 22080 . . . . . . . . . . . . 13  |-  ( ( ( y  i^i  ran  F )  e.  Fin  /\  A. x  e.  ( y  i^i  ran  F )
( `' F " { x } )  e.  dom  vol )  ->  U_ x  e.  ( y  i^i  ran  F
) ( `' F " { x } )  e.  dom  vol )
5438, 52, 53syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
5533, 54eqeltrrd 2546 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y )  e.  dom  vol )
5655ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol ) )
5756alrimiv 1720 . . . . . . . . 9  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
5857ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
59 elndif 3624 . . . . . . . . 9  |-  ( 0  e.  x  ->  -.  0  e.  ( RR  \  x ) )
6059adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  -.  0  e.  ( RR  \  x ) )
61 reex 9600 . . . . . . . . . 10  |-  RR  e.  _V
62 difexg 4604 . . . . . . . . . 10  |-  ( RR  e.  _V  ->  ( RR  \  x )  e. 
_V )
6361, 62ax-mp 5 . . . . . . . . 9  |-  ( RR 
\  x )  e. 
_V
64 eleq2 2530 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  (
0  e.  y  <->  0  e.  ( RR  \  x
) ) )
6564notbid 294 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR 
\  x ) ) )
66 imaeq2 5343 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  ( `' F " y )  =  ( `' F " ( RR  \  x
) ) )
6766eleq1d 2526 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
)
6865, 67imbi12d 320 . . . . . . . . 9  |-  ( y  =  ( RR  \  x )  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  ( RR  \  x )  ->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
) )
6963, 68spcv 3200 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  ( RR  \  x
)  ->  ( `' F " ( RR  \  x ) )  e. 
dom  vol ) )
7058, 60, 69sylc 60 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  x
) )  e.  dom  vol )
71 difmbl 22079 . . . . . . 7  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( RR  \  x
) )  e.  dom  vol )  ->  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) )  e. 
dom  vol )
7220, 70, 71syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  e.  dom  vol )
7316, 72eqeltrrd 2546 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
x )  e.  dom  vol )
74 eleq2 2530 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
0  e.  y  <->  0  e.  x ) )
7574notbid 294 . . . . . . . . . 10  |-  ( y  =  x  ->  ( -.  0  e.  y  <->  -.  0  e.  x ) )
76 imaeq2 5343 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( `' F " y )  =  ( `' F " x ) )
7776eleq1d 2526 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " x )  e.  dom  vol )
)
7875, 77imbi12d 320 . . . . . . . . 9  |-  ( y  =  x  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  x  ->  ( `' F " x )  e.  dom  vol )
) )
7978spv 2012 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8057, 79syl 16 . . . . . . 7  |-  ( ph  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8180imp 429 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  x )  ->  ( `' F " x )  e.  dom  vol )
8281adantlr 714 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  -.  0  e.  x
)  ->  ( `' F " x )  e. 
dom  vol )
8373, 82pm2.61dan 791 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
8483ralrimiva 2871 . . 3  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
85 ismbf 22163 . . . 4  |-  ( F : RR --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
861, 85syl 16 . . 3  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
8784, 86mpbird 232 . 2  |-  ( ph  ->  F  e. MblFn )
88 mblvol 22067 . . . . . . . 8  |-  ( ( `' F " y )  e.  dom  vol  ->  ( vol `  ( `' F " y ) )  =  ( vol* `  ( `' F " y ) ) )
8955, 88syl 16 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  =  ( vol* `  ( `' F " y ) ) )
90 mblss 22068 . . . . . . . . 9  |-  ( ( `' F " y )  e.  dom  vol  ->  ( `' F " y ) 
C_  RR )
9155, 90syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y ) 
C_  RR )
92 mblvol 22067 . . . . . . . . . . 11  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( vol `  ( `' F " { x } ) )  =  ( vol* `  ( `' F " { x } ) ) )
9351, 92syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol* `  ( `' F " { x }
) ) )
94 i1fd.4 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9539, 49, 94syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9693, 95eqeltrrd 2546 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol* `  ( `' F " { x } ) )  e.  RR )
9738, 96fsumrecl 13568 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) )  e.  RR )
9833fveq2d 5876 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  =  ( vol* `  ( `' F " y ) ) )
99 mblss 22068 . . . . . . . . . . . . 13  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( `' F " { x } ) 
C_  RR )
10051, 99syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  C_  RR )
101100, 96jca 532 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( ( `' F " { x } )  C_  RR  /\  ( vol* `  ( `' F " { x } ) )  e.  RR ) )
102101ralrimiva 2871 . . . . . . . . . 10  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( ( `' F " { x } )  C_  RR  /\  ( vol* `  ( `' F " { x } ) )  e.  RR ) )
103 ovolfiniun 22038 . . . . . . . . . 10  |-  ( ( ( y  i^i  ran  F )  e.  Fin  /\  A. x  e.  ( y  i^i  ran  F )
( ( `' F " { x } ) 
C_  RR  /\  ( vol* `  ( `' F " { x } ) )  e.  RR ) )  -> 
( vol* `  U_ x  e.  ( y  i^i  ran  F )
( `' F " { x } ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) ) )
10438, 102, 103syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) ) )
10598, 104eqbrtrrd 4478 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) ) )
106 ovollecl 22020 . . . . . . . 8  |-  ( ( ( `' F "
y )  C_  RR  /\ 
sum_ x  e.  (
y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) )  e.  RR  /\  ( vol* `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F
) ( vol* `  ( `' F " { x } ) ) )  ->  ( vol* `  ( `' F " y ) )  e.  RR )
10791, 97, 105, 106syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  ( `' F " y ) )  e.  RR )
10889, 107eqeltrd 2545 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  e.  RR )
109108ex 434 . . . . 5  |-  ( ph  ->  ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR ) )
110109alrimiv 1720 . . . 4  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR ) )
111 neldifsn 4159 . . . 4  |-  -.  0  e.  ( RR  \  {
0 } )
112 difexg 4604 . . . . . 6  |-  ( RR  e.  _V  ->  ( RR  \  { 0 } )  e.  _V )
11361, 112ax-mp 5 . . . . 5  |-  ( RR 
\  { 0 } )  e.  _V
114 eleq2 2530 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( 0  e.  y  <->  0  e.  ( RR  \  { 0 } ) ) )
115114notbid 294 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR  \  { 0 } ) ) )
116 imaeq2 5343 . . . . . . . 8  |-  ( y  =  ( RR  \  { 0 } )  ->  ( `' F " y )  =  ( `' F " ( RR 
\  { 0 } ) ) )
117116fveq2d 5876 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( vol `  ( `' F " y ) )  =  ( vol `  ( `' F "
( RR  \  {
0 } ) ) ) )
118117eleq1d 2526 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( vol `  ( `' F "
y ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
119115, 118imbi12d 320 . . . . 5  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR )  <-> 
( -.  0  e.  ( RR  \  {
0 } )  -> 
( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) ) )
120113, 119spcv 3200 . . . 4  |-  ( A. y ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR )  ->  ( -.  0  e.  ( RR  \  { 0 } )  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
121110, 111, 120mpisyl 18 . . 3  |-  ( ph  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
1221, 34, 1213jca 1176 . 2  |-  ( ph  ->  ( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
123 isi1f 22207 . 2  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
12487, 122, 123sylanbrc 664 1  |-  ( ph  ->  F  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U_ciun 4332   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594   Fincfn 7535   RRcr 9508   0cc0 9509   RR*cxr 9644    <_ cle 9646   (,)cioo 11554   sum_csu 13520   vol*covol 22000   volcvol 22001  MblFncmbf 22149   S.1citg1 22150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xadd 11344  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-xmet 18539  df-met 18540  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155
This theorem is referenced by:  i1f0  22220  i1f1  22223  i1fadd  22228  i1fmul  22229  i1fmulc  22236  i1fres  22238  mbfi1fseqlem4  22251  itg2addnclem2  30272  ftc1anclem3  30297
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