Users' Mathboxes Mathbox for Drahflow < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfrtrcl2 Structured version   Unicode version

Theorem dfrtrcl2 25148
Description: The two definitions  t * and  t *rec of the reflexive, transitive closure coincide if  R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
drrtrcl2.1  |-  ( ph  ->  Rel  R )
drrtrcl2.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
dfrtrcl2  |-  ( ph  ->  ( t * `  R )  =  ( t *rec `  R
) )

Proof of Theorem dfrtrcl2
Dummy variables  x  z  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2437 . . . 4  |-  ( ph  ->  ( x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } )  =  ( x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) )
2 dmeq 5070 . . . . . . . . . . 11  |-  ( x  =  R  ->  dom  x  =  dom  R )
3 rneq 5095 . . . . . . . . . . 11  |-  ( x  =  R  ->  ran  x  =  ran  R )
42, 3uneq12d 3502 . . . . . . . . . 10  |-  ( x  =  R  ->  ( dom  x  u.  ran  x
)  =  ( dom 
R  u.  ran  R
) )
54reseq2d 5146 . . . . . . . . 9  |-  ( x  =  R  ->  (  _I  |`  ( dom  x  u.  ran  x ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
65sseq1d 3375 . . . . . . . 8  |-  ( x  =  R  ->  (
(  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z )
)
7 id 20 . . . . . . . . 9  |-  ( x  =  R  ->  x  =  R )
87sseq1d 3375 . . . . . . . 8  |-  ( x  =  R  ->  (
x  C_  z  <->  R  C_  z
) )
96, 83anbi12d 1255 . . . . . . 7  |-  ( x  =  R  ->  (
( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )
) )
109abbidv 2550 . . . . . 6  |-  ( x  =  R  ->  { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
1110inteqd 4055 . . . . 5  |-  ( x  =  R  ->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
1211adantl 453 . . . 4  |-  ( (
ph  /\  x  =  R )  ->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
13 drrtrcl2.2 . . . 4  |-  ( ph  ->  R  e.  _V )
14 drrtrcl2.1 . . . . . . . . . 10  |-  ( ph  ->  Rel  R )
15 relfld 5395 . . . . . . . . . 10  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
1614, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
1716eqcomd 2441 . . . . . . . 8  |-  ( ph  ->  ( dom  R  u.  ran  R )  =  U. U. R )
1814, 13rtrclreclem.refl 25144 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  U. U. R )  C_  (
t *rec `  R
) )
19 id 20 . . . . . . . . . . 11  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( dom  R  u.  ran  R )  = 
U. U. R )
2019reseq2d 5146 . . . . . . . . . 10  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  =  (  _I  |`  U. U. R
) )
2120sseq1d 3375 . . . . . . . . 9  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t *rec `  R )  <->  (  _I  |` 
U. U. R )  C_  ( t *rec `  R ) ) )
2218, 21syl5ibr 213 . . . . . . . 8  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t *rec `  R )
) )
2317, 22mpcom 34 . . . . . . 7  |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
) )
2414, 13rtrclreclem.subset 25145 . . . . . . 7  |-  ( ph  ->  R  C_  ( t *rec `  R )
)
2514, 13rtrclreclem.trans 25146 . . . . . . 7  |-  ( ph  ->  ( ( t *rec
`  R )  o.  ( t *rec `  R ) )  C_  ( t *rec `  R ) )
26 fvex 5742 . . . . . . . 8  |-  ( t *rec `  R )  e.  _V
27 sseq2 3370 . . . . . . . . . . 11  |-  ( z  =  ( t *rec
`  R )  -> 
( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t *rec `  R )
) )
28 sseq2 3370 . . . . . . . . . . 11  |-  ( z  =  ( t *rec
`  R )  -> 
( R  C_  z  <->  R 
C_  ( t *rec
`  R ) ) )
29 id 20 . . . . . . . . . . . . 13  |-  ( z  =  ( t *rec
`  R )  -> 
z  =  ( t *rec `  R )
)
3029, 29coeq12d 5037 . . . . . . . . . . . 12  |-  ( z  =  ( t *rec
`  R )  -> 
( z  o.  z
)  =  ( ( t *rec `  R
)  o.  ( t *rec `  R )
) )
3130, 29sseq12d 3377 . . . . . . . . . . 11  |-  ( z  =  ( t *rec
`  R )  -> 
( ( z  o.  z )  C_  z  <->  ( ( t *rec `  R )  o.  (
t *rec `  R
) )  C_  (
t *rec `  R
) ) )
3227, 28, 313anbi123d 1254 . . . . . . . . . 10  |-  ( z  =  ( t *rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) )
3332a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( z  =  ( t *rec `  R
)  ->  ( (
(  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) ) )
3433alrimiv 1641 . . . . . . . 8  |-  ( ph  ->  A. z ( z  =  ( t *rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) ) )
35 elabgt 3079 . . . . . . . 8  |-  ( ( ( t *rec `  R )  e.  _V  /\ 
A. z ( z  =  ( t *rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) ) )  ->  ( ( t *rec `  R )  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) )
3626, 34, 35sylancr 645 . . . . . . 7  |-  ( ph  ->  ( ( t *rec
`  R )  e. 
{ z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t *rec `  R
)  /\  R  C_  (
t *rec `  R
)  /\  ( (
t *rec `  R
)  o.  ( t *rec `  R )
)  C_  ( t *rec `  R )
) ) )
3723, 24, 25, 36mpbir3and 1137 . . . . . 6  |-  ( ph  ->  ( t *rec `  R )  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) } )
38 ne0i 3634 . . . . . 6  |-  ( ( t *rec `  R
)  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  ->  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =/=  (/) )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =/=  (/) )
40 intex 4356 . . . . 5  |-  ( { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  =/=  (/)  <->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  e.  _V )
4139, 40sylib 189 . . . 4  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  e.  _V )
421, 12, 13, 41fvmptd 5810 . . 3  |-  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
43 intss1 4065 . . . . 5  |-  ( ( t *rec `  R
)  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  C_  ( t *rec
`  R ) )
4437, 43syl 16 . . . 4  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  C_  ( t *rec
`  R ) )
45 vex 2959 . . . . . . . 8  |-  s  e. 
_V
46 sseq2 3370 . . . . . . . . 9  |-  ( z  =  s  ->  (
(  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  s )
)
47 sseq2 3370 . . . . . . . . 9  |-  ( z  =  s  ->  ( R  C_  z  <->  R  C_  s
) )
48 id 20 . . . . . . . . . . 11  |-  ( z  =  s  ->  z  =  s )
4948, 48coeq12d 5037 . . . . . . . . . 10  |-  ( z  =  s  ->  (
z  o.  z )  =  ( s  o.  s ) )
5049, 48sseq12d 3377 . . . . . . . . 9  |-  ( z  =  s  ->  (
( z  o.  z
)  C_  z  <->  ( s  o.  s )  C_  s
) )
5146, 47, 503anbi123d 1254 . . . . . . . 8  |-  ( z  =  s  ->  (
( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )
) )
5245, 51elab 3082 . . . . . . 7  |-  ( s  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )
)
5314, 13rtrclreclem.min 25147 . . . . . . . 8  |-  ( ph  ->  A. s ( ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )  ->  ( t *rec `  R )  C_  s
) )
545319.21bi 1774 . . . . . . 7  |-  ( ph  ->  ( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )  ->  ( t *rec `  R )  C_  s
) )
5552, 54syl5bi 209 . . . . . 6  |-  ( ph  ->  ( s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ->  ( t *rec
`  R )  C_  s ) )
5655ralrimiv 2788 . . . . 5  |-  ( ph  ->  A. s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ( t *rec `  R )  C_  s
)
57 ssint 4066 . . . . 5  |-  ( ( t *rec `  R
)  C_  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<-> 
A. s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ( t *rec `  R )  C_  s
)
5856, 57sylibr 204 . . . 4  |-  ( ph  ->  ( t *rec `  R )  C_  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
5944, 58eqssd 3365 . . 3  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =  ( t *rec `  R )
)
6042, 59eqtrd 2468 . 2  |-  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t *rec `  R )
)
61 df-rtrcl 21773 . . 3  |-  t
*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )
62 fveq1 5727 . . . . 5  |-  ( t *  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( t * `
 R )  =  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
) )
6362eqeq1d 2444 . . . 4  |-  ( t *  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( ( t * `  R )  =  ( t *rec
`  R )  <->  ( (
x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t *rec `  R )
) )
6463imbi2d 308 . . 3  |-  ( t *  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( ( ph  ->  ( t * `  R )  =  ( t *rec `  R
) )  <->  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t *rec `  R )
) ) )
6561, 64ax-mp 8 . 2  |-  ( (
ph  ->  ( t * `
 R )  =  ( t *rec `  R ) )  <->  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t *rec `  R )
) )
6660, 65mpbir 201 1  |-  ( ph  ->  ( t * `  R )  =  ( t *rec `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   _Vcvv 2956    u. cun 3318    C_ wss 3320   (/)c0 3628   U.cuni 4015   |^|cint 4050    e. cmpt 4266    _I cid 4493   dom cdm 4878   ran crn 4879    |` cres 4880    o. ccom 4882   Rel wrel 4883   ` cfv 5454   t *crtcl 21771   t *reccrtrcl 25141
This theorem is referenced by:  rtrclind  25149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-rtrcl 21773  df-relexp 25128  df-rtrclrec 25142
  Copyright terms: Public domain W3C validator