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Theorem i1fres 19598
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 19569 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21adantr 453 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F : RR --> RR )
3 ffn 5592 . . . . . . 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 5868 . . . . . 6  |-  ( ( F  Fn  RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  ran  F
)
64, 5sylan 459 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  ( F `  x )  e.  ran  F )
7 i1f0rn 19575 . . . . . 6  |-  ( F  e.  dom  S.1  ->  0  e.  ran  F )
87ad2antrr 708 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  0  e.  ran  F )
9 ifcl 3776 . . . . 5  |-  ( ( ( F `  x
)  e.  ran  F  /\  0  e.  ran  F )  ->  if (
x  e.  A , 
( F `  x
) ,  0 )  e.  ran  F )
106, 8, 9syl2anc 644 . . . 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 5894 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> ran  F )
13 frn 5598 . . . 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 5600 . . 3  |-  ( ( G : RR --> ran  F  /\  ran  F  C_  RR )  ->  G : RR --> RR )
1612, 14, 15syl2anc 644 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> RR )
17 i1frn 19570 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
1817adantr 453 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  e.  Fin )
19 frn 5598 . . . 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 7330 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  C_  ran  F
)  ->  ran  G  e. 
Fin )
2218, 20, 21syl2anc 644 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  e.  Fin )
23 eleq1 2497 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
24 fveq2 5729 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
25 eqidd 2438 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  0  =  0 )
2623, 24, 25ifbieq12d 3762 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  =  if ( z  e.  A , 
( F `  z
) ,  0 ) )
27 fvex 5743 . . . . . . . . . . . . . 14  |-  ( F `
 z )  e. 
_V
28 c0ex 9086 . . . . . . . . . . . . . 14  |-  0  e.  _V
2927, 28ifex 3798 . . . . . . . . . . . . 13  |-  if ( z  e.  A , 
( F `  z
) ,  0 )  e.  _V
3026, 11, 29fvmpt 5807 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3130adantl 454 . . . . . . . . . . 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 2445 . . . . . . . . . 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 3929 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  ->  y  =/=  0
)
3433ad2antlr 709 . . . . . . . . . . . . . 14  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  y  =/=  0 )
3534necomd 2688 . . . . . . . . . . . . 13  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  0  =/=  y )
36 iffalse 3747 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  0 )
3736neeq1d 2615 . . . . . . . . . . . . 13  |-  ( -.  z  e.  A  -> 
( if ( z  e.  A ,  ( F `  z ) ,  0 )  =/=  y  <->  0  =/=  y
) )
3835, 37syl5ibrcom 215 . . . . . . . . . . . 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 2667 . . . . . . . . . . 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 618 . . . . . . . . . 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 246 . . . . . . . . 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 3746 . . . . . . . . . . 11  |-  ( z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  ( F `
 z ) )
4342eqeq1d 2445 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( F `  z )  =  y ) )
4443pm5.32i 620 . . . . . . . . 9  |-  ( ( z  e.  A  /\  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y )  <-> 
( z  e.  A  /\  ( F `  z
)  =  y ) )
4541, 44syl6bb 254 . . . . . . . 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 624 . . . . . . 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 774 . . . . . . 7  |-  ( ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z
)  =  y ) )  <->  ( z  e.  A  /\  ( z  e.  RR  /\  ( F `  z )  =  y ) ) )
4846, 47syl6bb 254 . . . . . 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 5592 . . . . . . . . 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 453 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  G  Fn  RR )
52 fniniseg 5852 . . . . . . 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 453 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  F  Fn  RR )
55 fniniseg 5852 . . . . . . . 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 686 . . . . . 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 278 . . . . 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 3531 . . . . 5  |-  ( z  e.  ( A  i^i  ( `' F " { y } ) )  <->  ( z  e.  A  /\  z  e.  ( `' F " { y } ) ) )
6058, 59syl6bbr 256 . . . 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 2435 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  =  ( A  i^i  ( `' F " { y } ) ) )
62 simplr 733 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  A  e.  dom  vol )
63 i1fima 19571 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
6463ad2antrr 708 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
65 inmbl 19437 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( `' F " { y } )  e.  dom  vol )  ->  ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol )
6662, 64, 65syl2anc 644 . . 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 2511 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  e.  dom  vol )
6861fveq2d 5733 . . . 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 19427 . . . . 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 2469 . . 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 3563 . . . . 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 19428 . . . . 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 19427 . . . . . 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 19573 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' F " { y } ) )  e.  RR )
7978adantlr 697 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
8077, 79eqeltrrd 2512 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( `' F " { y } ) )  e.  RR )
81 ovolsscl 19383 . . . 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 1185 . . 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 2511 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  e.  RR )
8416, 22, 67, 83i1fd 19574 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600    \ cdif 3318    i^i cin 3320    C_ wss 3321   ifcif 3740   {csn 3815    e. cmpt 4267   `'ccnv 4878   dom cdm 4879   ran crn 4880   "cima 4882    Fn wfn 5450   -->wf 5451   ` cfv 5455   Fincfn 7110   RRcr 8990   0cc0 8991   vol *covol 19360   volcvol 19361   S.1citg1 19508
This theorem is referenced by:  i1fpos  19599  itg1climres  19607  itg2uba  19636  itg2splitlem  19641  itg2monolem1  19643  ftc1anclem5  26285  ftc1anclem7  26287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-sum 12481  df-rest 13651  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-top 16964  df-bases 16966  df-topon 16967  df-cmp 17451  df-ovol 19362  df-vol 19363  df-mbf 19513  df-itg1 19514
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