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Definition df-relexp 13084
Description: Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.)
Assertion
Ref Expression
df-relexp  |- ^r 
=  ( r  e. 
_V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )
Distinct variable group:    n, r, x, y, z

Detailed syntax breakdown of Definition df-relexp
StepHypRef Expression
1 crelexp 13083 . 2  class ^r
2 vr . . 3  setvar  r
3 vn . . 3  setvar  n
4 cvv 3080 . . 3  class  _V
5 cn0 10876 . . 3  class  NN0
63cv 1436 . . . . 5  class  n
7 cc0 9546 . . . . 5  class  0
86, 7wceq 1437 . . . 4  wff  n  =  0
9 cid 4763 . . . . 5  class  _I
102cv 1436 . . . . . . 7  class  r
1110cdm 4853 . . . . . 6  class  dom  r
1210crn 4854 . . . . . 6  class  ran  r
1311, 12cun 3434 . . . . 5  class  ( dom  r  u.  ran  r
)
149, 13cres 4855 . . . 4  class  (  _I  |`  ( dom  r  u. 
ran  r ) )
15 vx . . . . . . 7  setvar  x
16 vy . . . . . . 7  setvar  y
1715cv 1436 . . . . . . . 8  class  x
1817, 10ccom 4857 . . . . . . 7  class  ( x  o.  r )
1915, 16, 4, 4, 18cmpt2 6307 . . . . . 6  class  ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) )
20 vz . . . . . . 7  setvar  z
2120, 4, 10cmpt 4482 . . . . . 6  class  ( z  e.  _V  |->  r )
22 c1 9547 . . . . . 6  class  1
2319, 21, 22cseq 12219 . . . . 5  class  seq 1
( ( x  e. 
_V ,  y  e. 
_V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  r ) )
246, 23cfv 5601 . . . 4  class  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n )
258, 14, 24cif 3911 . . 3  class  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) )
262, 3, 4, 5, 25cmpt2 6307 . 2  class  ( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )
271, 26wceq 1437 1  wff ^r 
=  ( r  e. 
_V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  relexp0g  13085  relexpsucnnr  13088  relexp1g  13089
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