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Theorem heiborlem3 26640
Description: Lemma for heibor 26648. Using countable choice ax-cc 8077, we have fixed in advance a collection of finite  2 ^ -u n nets  ( F `  n ) for  X (note that an  r-net is a set of points in  X whose  r -balls cover  X). The set  G is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set  K). If the theorem was false, then  X would be in  K, and so some ball at each level would also be in  K. But we can say more than this; given a ball 
( y B n ) on level  n, since level  n  +  1 covers the space and thus also  (
y B n ), using heiborlem1 26638 there is a ball on the next level whose intersection with  ( y B n ) also has no finite subcover. Now since the set 
G is a countable union of finite sets, it is countable (which needs ax-cc 8077 via iunctb 8212), and so we can apply ax-cc 8077 to  G directly to get a function from  G to itself, which points from each ball in  K to a ball on the next level in  K, and such that the intersection between these balls is also in  K. (Contributed by Jeff Madsen, 18-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
Assertion
Ref Expression
heiborlem3  |-  ( ph  ->  E. g A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
Distinct variable groups:    x, n, y, u, F    x, g, G    ph, g, x    g, m, n, u, v, y, z, D, x    B, g, n, u, v, y   
g, J, m, n, u, v, x, y, z    U, g, n, u, v, x, y, z   
g, X, m, n, u, v, x, y, z    g, K, n, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    B( z, m)    U( m)    F( z, v, g, m)    G( y, z, v, u, m, n)    K( v, u, m)

Proof of Theorem heiborlem3
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 nn0ex 9987 . . . . . 6  |-  NN0  e.  _V
2 fvex 5555 . . . . . . 7  |-  ( F `
 t )  e. 
_V
3 snex 4232 . . . . . . 7  |-  { t }  e.  _V
42, 3xpex 4817 . . . . . 6  |-  ( ( F `  t )  X.  { t } )  e.  _V
51, 4iunex 5786 . . . . 5  |-  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  e.  _V
6 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
76relopabi 4827 . . . . . . . 8  |-  Rel  G
8 1st2nd 6182 . . . . . . . 8  |-  ( ( Rel  G  /\  x  e.  G )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
97, 8mpan 651 . . . . . . 7  |-  ( x  e.  G  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
109eleq1d 2362 . . . . . . . . . . 11  |-  ( x  e.  G  ->  (
x  e.  G  <->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  G ) )
11 df-br 4040 . . . . . . . . . . 11  |-  ( ( 1st `  x ) G ( 2nd `  x
)  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  G
)
1210, 11syl6bbr 254 . . . . . . . . . 10  |-  ( x  e.  G  ->  (
x  e.  G  <->  ( 1st `  x ) G ( 2nd `  x ) ) )
13 heibor.1 . . . . . . . . . . 11  |-  J  =  ( MetOpen `  D )
14 heibor.3 . . . . . . . . . . 11  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
15 fvex 5555 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
16 fvex 5555 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
1713, 14, 6, 15, 16heiborlem2 26639 . . . . . . . . . 10  |-  ( ( 1st `  x ) G ( 2nd `  x
)  <->  ( ( 2nd `  x )  e.  NN0  /\  ( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) )  /\  (
( 1st `  x
) B ( 2nd `  x ) )  e.  K ) )
1812, 17syl6bb 252 . . . . . . . . 9  |-  ( x  e.  G  ->  (
x  e.  G  <->  ( ( 2nd `  x )  e. 
NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x
) B ( 2nd `  x ) )  e.  K ) ) )
1918ibi 232 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x ) B ( 2nd `  x
) )  e.  K
) )
2016snid 3680 . . . . . . . . . . . 12  |-  ( 2nd `  x )  e.  {
( 2nd `  x
) }
21 opelxp 4735 . . . . . . . . . . . 12  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } )  <-> 
( ( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) )  /\  ( 2nd `  x )  e. 
{ ( 2nd `  x
) } ) )
2220, 21mpbiran2 885 . . . . . . . . . . 11  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } )  <-> 
( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) ) )
23 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  ( F `  t )  =  ( F `  ( 2nd `  x ) ) )
24 sneq 3664 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  { t }  =  { ( 2nd `  x ) } )
2523, 24xpeq12d 4730 . . . . . . . . . . . . 13  |-  ( t  =  ( 2nd `  x
)  ->  ( ( F `  t )  X.  { t } )  =  ( ( F `
 ( 2nd `  x
) )  X.  {
( 2nd `  x
) } ) )
2625eleq2d 2363 . . . . . . . . . . . 12  |-  ( t  =  ( 2nd `  x
)  ->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  ( ( F `  t )  X.  { t } )  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } ) ) )
2726rspcev 2897 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } ) )  ->  E. t  e.  NN0  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
2822, 27sylan2br 462 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )  ->  E. t  e.  NN0  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
29 eliun 3925 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  U_ t  e.  NN0  ( ( F `  t )  X.  { t } )  <->  E. t  e.  NN0  <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
3028, 29sylibr 203 . . . . . . . . 9  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) )
31303adant3 975 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x ) B ( 2nd `  x
) )  e.  K
)  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  U_ t  e. 
NN0  ( ( F `
 t )  X. 
{ t } ) )
3219, 31syl 15 . . . . . . 7  |-  ( x  e.  G  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) )
339, 32eqeltrd 2370 . . . . . 6  |-  ( x  e.  G  ->  x  e.  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } ) )
3433ssriv 3197 . . . . 5  |-  G  C_  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )
35 ssdomg 6923 . . . . 5  |-  ( U_ t  e.  NN0  ( ( F `  t )  X.  { t } )  e.  _V  ->  ( G  C_  U_ t  e. 
NN0  ( ( F `
 t )  X. 
{ t } )  ->  G  ~<_  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) ) )
365, 34, 35mp2 17 . . . 4  |-  G  ~<_  U_ t  e.  NN0  ( ( F `  t )  X.  { t } )
37 nn0ennn 11057 . . . . . . 7  |-  NN0  ~~  NN
38 nnenom 11058 . . . . . . 7  |-  NN  ~~  om
3937, 38entri 6931 . . . . . 6  |-  NN0  ~~  om
40 endom 6904 . . . . . 6  |-  ( NN0  ~~  om  ->  NN0  ~<_  om )
4139, 40ax-mp 8 . . . . 5  |-  NN0  ~<_  om
42 vex 2804 . . . . . . . 8  |-  t  e. 
_V
432, 42xpsnen 6962 . . . . . . 7  |-  ( ( F `  t )  X.  { t } )  ~~  ( F `
 t )
44 inss2 3403 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
Fin
45 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
46 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  t  e.  NN0 )  ->  ( F `  t )  e.  ( ~P X  i^i  Fin ) )
4745, 46sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  ( ~P X  i^i  Fin ) )
4844, 47sseldi 3191 . . . . . . . 8  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  Fin )
49 isfinite 7369 . . . . . . . . 9  |-  ( ( F `  t )  e.  Fin  <->  ( F `  t )  ~<  om )
50 sdomdom 6905 . . . . . . . . 9  |-  ( ( F `  t ) 
~<  om  ->  ( F `  t )  ~<_  om )
5149, 50sylbi 187 . . . . . . . 8  |-  ( ( F `  t )  e.  Fin  ->  ( F `  t )  ~<_  om )
5248, 51syl 15 . . . . . . 7  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  ~<_  om )
53 endomtr 6935 . . . . . . 7  |-  ( ( ( ( F `  t )  X.  {
t } )  ~~  ( F `  t )  /\  ( F `  t )  ~<_  om )  ->  ( ( F `  t )  X.  {
t } )  ~<_  om )
5443, 52, 53sylancr 644 . . . . . 6  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( ( F `  t )  X.  { t } )  ~<_  om )
5554ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
56 iunctb 8212 . . . . 5  |-  ( ( NN0  ~<_  om  /\  A. t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  ~<_  om )  ->  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  ~<_  om )
5741, 55, 56sylancr 644 . . . 4  |-  ( ph  ->  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
58 domtr 6930 . . . 4  |-  ( ( G  ~<_  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } )  /\  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )  ~<_  om )  ->  G  ~<_  om )
5936, 57, 58sylancr 644 . . 3  |-  ( ph  ->  G  ~<_  om )
6019simp1d 967 . . . . . . . . 9  |-  ( x  e.  G  ->  ( 2nd `  x )  e. 
NN0 )
61 peano2nn0 10020 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  NN0  ->  ( ( 2nd `  x )  +  1 )  e. 
NN0 )
6260, 61syl 15 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
63 ffvelrn 5679 . . . . . . . 8  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  ( ( 2nd `  x
)  +  1 )  e.  NN0 )  -> 
( F `  (
( 2nd `  x
)  +  1 ) )  e.  ( ~P X  i^i  Fin )
)
6445, 62, 63syl2an 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e.  ( ~P X  i^i  Fin ) )
6544, 64sseldi 3191 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
Fin )
66 iunin2 3982 . . . . . . . 8  |-  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( ( B `  x
)  i^i  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
67 heibor.8 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
68 oveq1 5881 . . . . . . . . . . . . . . . 16  |-  ( y  =  t  ->  (
y B n )  =  ( t B n ) )
6968cbviunv 3957 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  n ) ( t B n )
70 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( F `  n )  =  ( F `  ( ( 2nd `  x )  +  1 ) ) )
7170iuneq1d 3944 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ t  e.  ( F `  n
) ( t B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B n ) )
7269, 71syl5eq 2340 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B n ) )
73 oveq2 5882 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( t B n )  =  ( t B ( ( 2nd `  x
)  +  1 ) ) )
7473iuneq2d 3946 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7572, 74eqtrd 2328 . . . . . . . . . . . . 13  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7675eqeq2d 2307 . . . . . . . . . . . 12  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( X  =  U_ y  e.  ( F `  n ) ( y B n )  <->  X  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) ) )
7776rspccva 2896 . . . . . . . . . . 11  |-  ( ( A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n )  /\  (
( 2nd `  x
)  +  1 )  e.  NN0 )  ->  X  =  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7867, 62, 77syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  X  =  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7978ineq2d 3383 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  X )  =  ( ( B `
 x )  i^i  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( t B ( ( 2nd `  x )  +  1 ) ) ) )
809fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( x  e.  G  ->  ( B `  x )  =  ( B `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
81 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) B ( 2nd `  x
) )  =  ( B `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
8280, 81syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( x  e.  G  ->  ( B `  x )  =  ( ( 1st `  x ) B ( 2nd `  x ) ) )
8382adantl 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  =  ( ( 1st `  x ) B ( 2nd `  x ) ) )
84 inss1 3402 . . . . . . . . . . . . . . . 16  |-  ( ~P X  i^i  Fin )  C_ 
~P X
85 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  ( 2nd `  x
)  e.  NN0 )  ->  ( F `  ( 2nd `  x ) )  e.  ( ~P X  i^i  Fin ) )
8645, 60, 85syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e.  ( ~P X  i^i  Fin ) )
8784, 86sseldi 3191 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e. 
~P X )
88 elpwi 3646 . . . . . . . . . . . . . . 15  |-  ( ( F `  ( 2nd `  x ) )  e. 
~P X  ->  ( F `  ( 2nd `  x ) )  C_  X )
8987, 88syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  C_  X )
9019simp2d 968 . . . . . . . . . . . . . . 15  |-  ( x  e.  G  ->  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )
9190adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )
9289, 91sseldd 3194 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  X )
9360adantl 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 2nd `  x )  e. 
NN0 )
94 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( z  =  ( 1st `  x
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
95 oveq2 5882 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( 2nd `  x
)  ->  ( 2 ^ m )  =  ( 2 ^ ( 2nd `  x ) ) )
9695oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( m  =  ( 2nd `  x
)  ->  ( 1  /  ( 2 ^ m ) )  =  ( 1  /  (
2 ^ ( 2nd `  x ) ) ) )
9796oveq2d 5890 . . . . . . . . . . . . . 14  |-  ( m  =  ( 2nd `  x
)  ->  ( ( 1st `  x ) (
ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
98 heibor.5 . . . . . . . . . . . . . 14  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
99 ovex 5899 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  e.  _V
10094, 97, 98, 99ovmpt2 5999 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  e.  X  /\  ( 2nd `  x )  e.  NN0 )  -> 
( ( 1st `  x
) B ( 2nd `  x ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
10192, 93, 100syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
( 1st `  x
) B ( 2nd `  x ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
10283, 101eqtrd 2328 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  =  ( ( 1st `  x ) ( ball `  D ) ( 1  /  ( 2 ^ ( 2nd `  x
) ) ) ) )
103 heibor.6 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( CMet `  X ) )
104 cmetmet 18728 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
105103, 104syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  ( Met `  X ) )
106 metxmet 17915 . . . . . . . . . . . . . 14  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
107105, 106syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  D  e.  ( * Met `  X ) )
108107adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  D  e.  ( * Met `  X
) )
109 2nn 9893 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
110 nnexpcl 11132 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  NN  /\  ( 2nd `  x )  e.  NN0 )  -> 
( 2 ^ ( 2nd `  x ) )  e.  NN )
111109, 93, 110sylancr 644 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  NN )
112111nnrpd 10405 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  RR+ )
113112rpreccld 10416 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e.  RR+ )
114113rpxrd 10407 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e. 
RR* )
115 blssm 17984 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  ( 1st `  x
)  e.  X  /\  ( 1  /  (
2 ^ ( 2nd `  x ) ) )  e.  RR* )  ->  (
( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  C_  X
)
116108, 92, 114, 115syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  (
( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  C_  X
)
117102, 116eqsstrd 3225 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_  X )
118 df-ss 3179 . . . . . . . . . 10  |-  ( ( B `  x ) 
C_  X  <->  ( ( B `  x )  i^i  X )  =  ( B `  x ) )
119117, 118sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  X )  =  ( B `  x ) )
12079, 119eqtr3d 2330 . . . . . . . 8  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x ) )
12166, 120syl5eq 2340 . . . . . . 7  |-  ( (
ph  /\  x  e.  G )  ->  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x ) )
122 eqimss2 3244 . . . . . . 7  |-  ( U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x )  ->  ( B `  x )  C_  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) ) )
123121, 122syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_ 
U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) ) )
12419simp3d 969 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 1st `  x
) B ( 2nd `  x ) )  e.  K )
12582, 124eqeltrd 2370 . . . . . . 7  |-  ( x  e.  G  ->  ( B `  x )  e.  K )
126125adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  e.  K )
127 fvex 5555 . . . . . . . 8  |-  ( B `
 x )  e. 
_V
128127inex1 4171 . . . . . . 7  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  _V
12913, 14, 128heiborlem1 26638 . . . . . 6  |-  ( ( ( F `  (
( 2nd `  x
)  +  1 ) )  e.  Fin  /\  ( B `  x ) 
C_  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  /\  ( B `
 x )  e.  K )  ->  E. t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)
13065, 123, 126, 129syl3anc 1182 . . . . 5  |-  ( (
ph  /\  x  e.  G )  ->  E. t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)
13184, 64sseldi 3191 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
~P X )
132 elpwi 3646 . . . . . . . . . . . 12  |-  ( ( F `  ( ( 2nd `  x )  +  1 ) )  e.  ~P X  -> 
( F `  (
( 2nd `  x
)  +  1 ) )  C_  X )
133131, 132syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  X )
13413mopnuni 18003 . . . . . . . . . . . . 13  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
135107, 134syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  X  =  U. J
)
136135adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  X  =  U. J )
137133, 136sseqtrd 3227 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  U. J )
138137sselda 3193 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  G )  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) )  ->  t  e.  U. J )
139138adantrr 697 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
t  e.  U. J
)
14062adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
141 id 19 . . . . . . . . . 10  |-  ( t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) )  ->  t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) )
142 snfi 6957 . . . . . . . . . . . 12  |-  { ( t B ( ( 2nd `  x )  +  1 ) ) }  e.  Fin
143 inss2 3403 . . . . . . . . . . . . 13  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  (
t B ( ( 2nd `  x )  +  1 ) )
144 ovex 5899 . . . . . . . . . . . . . . 15  |-  ( t B ( ( 2nd `  x )  +  1 ) )  e.  _V
145144unisn 3859 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  ( t B ( ( 2nd `  x )  +  1 ) )
146 uniiun 3971 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
147145, 146eqtr3i 2318 . . . . . . . . . . . . 13  |-  ( t B ( ( 2nd `  x )  +  1 ) )  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
148143, 147sseqtri 3223 . . . . . . . . . . . 12  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  U_ g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g
149 vex 2804 . . . . . . . . . . . . 13  |-  g  e. 
_V
15013, 14, 149heiborlem1 26638 . . . . . . . . . . . 12  |-  ( ( { ( t B ( ( 2nd `  x
)  +  1 ) ) }  e.  Fin  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  C_  U_ g  e. 
{ ( t B ( ( 2nd `  x
)  +  1 ) ) } g  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K )  ->  E. g  e.  {
( t B ( ( 2nd `  x
)  +  1 ) ) } g  e.  K )
151142, 148, 150mp3an12 1267 . . . . . . . . . . 11  |-  ( ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  E. g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g  e.  K )
152 eleq1 2356 . . . . . . . . . . . 12  |-  ( g  =  ( t B ( ( 2nd `  x
)  +  1 ) )  ->  ( g  e.  K  <->  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
153144, 152rexsn 3688 . . . . . . . . . . 11  |-  ( E. g  e.  { ( t B ( ( 2nd `  x )  +  1 ) ) } g  e.  K  <->  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K )
154151, 153sylib 188 . . . . . . . . . 10  |-  ( ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K
)
155 ovex 5899 . . . . . . . . . . . 12  |-  ( ( 2nd `  x )  +  1 )  e. 
_V
15613, 14, 6, 42, 155heiborlem2 26639 . . . . . . . . . . 11  |-  ( t G ( ( 2nd `  x )  +  1 )  <->  ( ( ( 2nd `  x )  +  1 )  e. 
NN0  /\  t  e.  ( F `  ( ( 2nd `  x )  +  1 ) )  /\  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
157156biimpri 197 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  x
)  +  1 )  e.  NN0  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) )  /\  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K
)  ->  t G
( ( 2nd `  x
)  +  1 ) )
158140, 141, 154, 157syl3an 1224 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  G )  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  ->  t G
( ( 2nd `  x
)  +  1 ) )
1591583expb 1152 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
t G ( ( 2nd `  x )  +  1 ) )
160 simprr 733 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K )
161139, 159, 160jca32 521 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
( t  e.  U. J  /\  ( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
162161ex 423 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  (
( t  e.  ( F `  ( ( 2nd `  x )  +  1 ) )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  ->  ( t  e.  U. J  /\  (
t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) ) ) )
163162reximdv2 2665 . . . . 5  |-  ( (
ph  /\  x  e.  G )  ->  ( E. t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
164130, 163mpd 14 . . . 4  |-  ( (
ph  /\  x  e.  G )  ->  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
165164ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  G  E. t  e.  U. J
( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
166 fvex 5555 . . . . . 6  |-  ( MetOpen `  D )  e.  _V
16713, 166eqeltri 2366 . . . . 5  |-  J  e. 
_V
168167uniex 4532 . . . 4  |-  U. J  e.  _V
169 breq1 4042 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
t G ( ( 2nd `  x )  +  1 )  <->  ( g `  x ) G ( ( 2nd `  x
)  +  1 ) ) )
170 oveq1 5881 . . . . . . 7  |-  ( t  =  ( g `  x )  ->  (
t B ( ( 2nd `  x )  +  1 ) )  =  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )
171170ineq2d 3383 . . . . . 6  |-  ( t  =  ( g `  x )  ->  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  =  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) ) )
172171eleq1d 2362 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  <->  ( ( B `  x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
173169, 172anbi12d 691 . . . 4  |-  ( t  =  ( g `  x )  ->  (
( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  <->  ( ( g `
 x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
174168, 173axcc4dom 8083 . . 3  |-  ( ( G  ~<_  om  /\  A. x  e.  G  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )  ->  E. g
( g : G --> U. J  /\  A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
17559, 165, 174syl2anc 642 . 2  |-  ( ph  ->  E. g ( g : G --> U. J  /\  A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
176 simpr 447 . . 3  |-  ( ( g : G --> U. J  /\  A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )  ->  A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
177176eximi 1566 . 2  |-  ( E. g ( g : G --> U. J  /\  A. x  e.  G  (
( g `  x
) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  ->  E. g A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
178175, 177syl 15 1  |-  ( ph  ->  E. g A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   {copab 4092   omcom 4672    X. cxp 4703   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879   1c1 8754    + caddc 8756   RR*cxr 8882    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ^cexp 11120   * Metcxmt 16385   Metcme 16386   ballcbl 16387   MetOpencmopn 16388   CMetcms 18696
This theorem is referenced by:  heiborlem10  26647
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-cc 8077  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
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-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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  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-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-seq 11063  df-exp 11121  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cmet 18699
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