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

Theorem dfrtrcl2 27484
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 2452 . . . 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 5138 . . . . . . . . . . 11  |-  ( x  =  R  ->  dom  x  =  dom  R )
3 rneq 5163 . . . . . . . . . . 11  |-  ( x  =  R  ->  ran  x  =  ran  R )
42, 3uneq12d 3609 . . . . . . . . . 10  |-  ( x  =  R  ->  ( dom  x  u.  ran  x
)  =  ( dom 
R  u.  ran  R
) )
54reseq2d 5208 . . . . . . . . 9  |-  ( x  =  R  ->  (  _I  |`  ( dom  x  u.  ran  x ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
65sseq1d 3481 . . . . . . . 8  |-  ( x  =  R  ->  (
(  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z )
)
7 id 22 . . . . . . . . 9  |-  ( x  =  R  ->  x  =  R )
87sseq1d 3481 . . . . . . . 8  |-  ( x  =  R  ->  (
x  C_  z  <->  R  C_  z
) )
96, 83anbi12d 1291 . . . . . . 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 2587 . . . . . 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 4231 . . . . 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 466 . . . 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 5461 . . . . . . . . . 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 2459 . . . . . . . 8  |-  ( ph  ->  ( dom  R  u.  ran  R )  =  U. U. R )
1814, 13rtrclreclem.refl 27480 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  U. U. R )  C_  (
t*rec `  R
) )
19 id 22 . . . . . . . . . . 11  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( dom  R  u.  ran  R )  = 
U. U. R )
2019reseq2d 5208 . . . . . . . . . 10  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  =  (  _I  |`  U. U. R
) )
2120sseq1d 3481 . . . . . . . . 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 221 . . . . . . . 8  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t*rec `  R )
) )
2317, 22mpcom 36 . . . . . . 7  |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
) )
2414, 13rtrclreclem.subset 27481 . . . . . . 7  |-  ( ph  ->  R  C_  ( t*rec `  R )
)
2514, 13rtrclreclem.trans 27482 . . . . . . 7  |-  ( ph  ->  ( ( t*rec
`  R )  o.  ( t*rec `  R ) )  C_  ( t*rec `  R ) )
26 fvex 5799 . . . . . . . 8  |-  ( t*rec `  R )  e.  _V
27 sseq2 3476 . . . . . . . . . . 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 3476 . . . . . . . . . . 11  |-  ( z  =  ( t*rec
`  R )  -> 
( R  C_  z  <->  R 
C_  ( t*rec
`  R ) ) )
29 id 22 . . . . . . . . . . . . 13  |-  ( z  =  ( t*rec
`  R )  -> 
z  =  ( t*rec `  R )
)
3029, 29coeq12d 5102 . . . . . . . . . . . 12  |-  ( z  =  ( t*rec
`  R )  -> 
( z  o.  z
)  =  ( ( t*rec `  R
)  o.  ( t*rec `  R )
) )
3130, 29sseq12d 3483 . . . . . . . . . . 11  |-  ( z  =  ( t*rec
`  R )  -> 
( ( z  o.  z )  C_  z  <->  ( ( t*rec `  R )  o.  (
t*rec `  R
) )  C_  (
t*rec `  R
) ) )
3227, 28, 313anbi123d 1290 . . . . . . . . . 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 1686 . . . . . . . 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 3200 . . . . . . . 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 663 . . . . . . 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 1171 . . . . . 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 3741 . . . . . 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 4546 . . . . 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 196 . . . 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 5878 . . 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 4241 . . . . 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 3071 . . . . . . . 8  |-  s  e. 
_V
46 sseq2 3476 . . . . . . . . 9  |-  ( z  =  s  ->  (
(  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  s )
)
47 sseq2 3476 . . . . . . . . 9  |-  ( z  =  s  ->  ( R  C_  z  <->  R  C_  s
) )
48 id 22 . . . . . . . . . . 11  |-  ( z  =  s  ->  z  =  s )
4948, 48coeq12d 5102 . . . . . . . . . 10  |-  ( z  =  s  ->  (
z  o.  z )  =  ( s  o.  s ) )
5049, 48sseq12d 3483 . . . . . . . . 9  |-  ( z  =  s  ->  (
( z  o.  z
)  C_  z  <->  ( s  o.  s )  C_  s
) )
5146, 47, 503anbi123d 1290 . . . . . . . 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 3203 . . . . . . 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 27483 . . . . . . . 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 1806 . . . . . . 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 217 . . . . . 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 2820 . . . . 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 4242 . . . . 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 212 . . . 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 3471 . . 3  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =  ( t*rec `  R )
)
6042, 59eqtrd 2492 . 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 23807 . . 3  |-  t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } )
62 fveq1 5788 . . . . 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 2453 . . . 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 316 . . 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 5 . 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 209 1  |-  ( ph  ->  ( t* `  R )  =  ( t*rec `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2436    =/= wne 2644   A.wral 2795   _Vcvv 3068    u. cun 3424    C_ wss 3426   (/)c0 3735   U.cuni 4189   |^|cint 4226    |-> cmpt 4448    _I cid 4729   dom cdm 4938   ran crn 4939    |` cres 4940    o. ccom 4942   Rel wrel 4943   ` cfv 5516   t*crtcl 23805   t*reccrtrcl 27477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-seq 11908  df-rtrcl 23807  df-relexp 27464  df-rtrclrec 27478
This theorem is referenced by:  rtrclind  27485
  Copyright terms: Public domain W3C validator