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Theorem i1fres 19464
Description: The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside  A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
Hypothesis
Ref Expression
i1fres.1  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
Assertion
Ref Expression
i1fres  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem i1fres
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1ff 19435 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21adantr 452 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F : RR --> RR )
3 ffn 5531 . . . . . . 7  |-  ( F : RR --> RR  ->  F  Fn  RR )
42, 3syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F  Fn  RR )
5 fnfvelrn 5806 . . . . . 6  |-  ( ( F  Fn  RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  ran  F
)
64, 5sylan 458 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  ( F `  x )  e.  ran  F )
7 i1f0rn 19441 . . . . . 6  |-  ( F  e.  dom  S.1  ->  0  e.  ran  F )
87ad2antrr 707 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  0  e.  ran  F )
9 ifcl 3718 . . . . 5  |-  ( ( ( F `  x
)  e.  ran  F  /\  0  e.  ran  F )  ->  if (
x  e.  A , 
( F `  x
) ,  0 )  e.  ran  F )
106, 8, 9syl2anc 643 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  e.  ran  F
)
11 i1fres.1 . . . 4  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
1210, 11fmptd 5832 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> ran  F )
13 frn 5537 . . . 4  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
142, 13syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  C_  RR )
15 fss 5539 . . 3  |-  ( ( G : RR --> ran  F  /\  ran  F  C_  RR )  ->  G : RR --> RR )
1612, 14, 15syl2anc 643 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> RR )
17 i1frn 19436 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
1817adantr 452 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  e.  Fin )
19 frn 5537 . . . 4  |-  ( G : RR --> ran  F  ->  ran  G  C_  ran  F )
2012, 19syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  C_  ran  F )
21 ssfi 7265 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  C_  ran  F
)  ->  ran  G  e. 
Fin )
2218, 20, 21syl2anc 643 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  e.  Fin )
23 eleq1 2447 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
24 fveq2 5668 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
25 eqidd 2388 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  0  =  0 )
2623, 24, 25ifbieq12d 3704 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  =  if ( z  e.  A , 
( F `  z
) ,  0 ) )
27 fvex 5682 . . . . . . . . . . . . . 14  |-  ( F `
 z )  e. 
_V
28 c0ex 9018 . . . . . . . . . . . . . 14  |-  0  e.  _V
2927, 28ifex 3740 . . . . . . . . . . . . 13  |-  if ( z  e.  A , 
( F `  z
) ,  0 )  e.  _V
3026, 11, 29fvmpt 5745 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3130adantl 453 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3231eqeq1d 2395 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  if (
z  e.  A , 
( F `  z
) ,  0 )  =  y ) )
33 eldifsni 3871 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  ->  y  =/=  0
)
3433ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  y  =/=  0 )
3534necomd 2633 . . . . . . . . . . . . 13  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  0  =/=  y )
36 iffalse 3689 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  0 )
3736neeq1d 2563 . . . . . . . . . . . . 13  |-  ( -.  z  e.  A  -> 
( if ( z  e.  A ,  ( F `  z ) ,  0 )  =/=  y  <->  0  =/=  y
) )
3835, 37syl5ibrcom 214 . . . . . . . . . . . 12  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `
 z ) ,  0 )  =/=  y
) )
3938necon4bd 2612 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  -> 
z  e.  A ) )
4039pm4.71rd 617 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
4132, 40bitrd 245 . . . . . . . . 9  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
42 iftrue 3688 . . . . . . . . . . 11  |-  ( z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  ( F `
 z ) )
4342eqeq1d 2395 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( F `  z )  =  y ) )
4443pm5.32i 619 . . . . . . . . 9  |-  ( ( z  e.  A  /\  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y )  <-> 
( z  e.  A  /\  ( F `  z
)  =  y ) )
4541, 44syl6bb 253 . . . . . . . 8  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  ( F `  z )  =  y ) ) )
4645pm5.32da 623 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z )  =  y ) ) ) )
47 an12 773 . . . . . . 7  |-  ( ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z
)  =  y ) )  <->  ( z  e.  A  /\  ( z  e.  RR  /\  ( F `  z )  =  y ) ) )
4846, 47syl6bb 253 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
49 ffn 5531 . . . . . . . . 9  |-  ( G : RR --> ran  F  ->  G  Fn  RR )
5012, 49syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  Fn  RR )
5150adantr 452 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  G  Fn  RR )
52 fniniseg 5790 . . . . . . 7  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
5351, 52syl 16 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
544adantr 452 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  F  Fn  RR )
55 fniniseg 5790 . . . . . . . 8  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { y } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  y ) ) )
5654, 55syl 16 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' F " { y } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  y ) ) )
5756anbi2d 685 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  A  /\  z  e.  ( `' F " { y } ) )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
5848, 53, 573bitr4d 277 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  A  /\  z  e.  ( `' F " { y } ) ) ) )
59 elin 3473 . . . . 5  |-  ( z  e.  ( A  i^i  ( `' F " { y } ) )  <->  ( z  e.  A  /\  z  e.  ( `' F " { y } ) ) )
6058, 59syl6bbr 255 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
z  e.  ( A  i^i  ( `' F " { y } ) ) ) )
6160eqrdv 2385 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  =  ( A  i^i  ( `' F " { y } ) ) )
62 simplr 732 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  A  e.  dom  vol )
63 i1fima 19437 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
6463ad2antrr 707 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
65 inmbl 19303 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( `' F " { y } )  e.  dom  vol )  ->  ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol )
6662, 64, 65syl2anc 643 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  e.  dom  vol )
6761, 66eqeltrd 2461 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  e.  dom  vol )
6861fveq2d 5672 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol `  ( A  i^i  ( `' F " { y } ) ) ) )
69 mblvol 19293 . . . . 5  |-  ( ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol * `  ( A  i^i  ( `' F " { y } ) ) ) )
7066, 69syl 16 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol * `  ( A  i^i  ( `' F " { y } ) ) ) )
7168, 70eqtrd 2419 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol
* `  ( A  i^i  ( `' F " { y } ) ) ) )
72 inss2 3505 . . . . 5  |-  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )
7372a1i 11 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } ) )
74 mblss 19294 . . . . 5  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7564, 74syl 16 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  C_  RR )
76 mblvol 19293 . . . . . 6  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( vol `  ( `' F " { y } ) )  =  ( vol * `  ( `' F " { y } ) ) )
7764, 76syl 16 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol
* `  ( `' F " { y } ) ) )
78 i1fima2sn 19439 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' F " { y } ) )  e.  RR )
7978adantlr 696 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
8077, 79eqeltrrd 2462 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( `' F " { y } ) )  e.  RR )
81 ovolsscl 19249 . . . 4  |-  ( ( ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )  /\  ( `' F " { y } )  C_  RR  /\  ( vol * `  ( `' F " { y } ) )  e.  RR )  ->  ( vol * `  ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8273, 75, 80, 81syl3anc 1184 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8371, 82eqeltrd 2461 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  e.  RR )
8416, 22, 67, 83i1fd 19440 1  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260    i^i cin 3262    C_ wss 3263   ifcif 3682   {csn 3757    e. cmpt 4207   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394   Fincfn 7045   RRcr 8922   0cc0 8923   vol *covol 19226   volcvol 19227   S.1citg1 19374
This theorem is referenced by:  i1fpos  19465  itg1climres  19473  itg2uba  19502  itg2splitlem  19507  itg2monolem1  19509
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-rest 13577  df-topgen 13594  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-top 16886  df-bases 16888  df-topon 16889  df-cmp 17372  df-ovol 19228  df-vol 19229  df-mbf 19379  df-itg1 19380
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