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Definition df-relexp 24024
Description: Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
Assertion
Ref Expression
df-relexp  |-  ^ r  =  ( r  e. 
_V ,  n  e. 
NN0  |->  (  seq  0
( ( x  e. 
_V ,  y  e. 
_V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r ) ) ) `
 n ) )
Distinct variable group:    n, r, x, y, z

Detailed syntax breakdown of Definition df-relexp
StepHypRef Expression
1 crelexp 24023 . 2  class  ^ r
2 vr . . 3  set  r
3 vn . . 3  set  n
4 cvv 2788 . . 3  class  _V
5 cn0 9965 . . 3  class  NN0
63cv 1622 . . . 4  class  n
7 vx . . . . . 6  set  x
8 vy . . . . . 6  set  y
97cv 1622 . . . . . . 7  class  x
102cv 1622 . . . . . . 7  class  r
119, 10ccom 4693 . . . . . 6  class  ( x  o.  r )
127, 8, 4, 4, 11cmpt2 5860 . . . . 5  class  ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) )
13 vz . . . . . 6  set  z
14 cid 4304 . . . . . . 7  class  _I
1510cuni 3827 . . . . . . . 8  class  U. r
1615cuni 3827 . . . . . . 7  class  U. U. r
1714, 16cres 4691 . . . . . 6  class  (  _I  |`  U. U. r )
1813, 4, 17cmpt 4077 . . . . 5  class  ( z  e.  _V  |->  (  _I  |`  U. U. r ) )
19 cc0 8737 . . . . 5  class  0
2012, 18, 19cseq 11046 . . . 4  class  seq  0
( ( x  e. 
_V ,  y  e. 
_V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r ) ) )
216, 20cfv 5255 . . 3  class  (  seq  0 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  (  _I  |`  U. U. r ) ) ) `
 n )
222, 3, 4, 5, 21cmpt2 5860 . 2  class  ( r  e.  _V ,  n  e.  NN0  |->  (  seq  0
( ( x  e. 
_V ,  y  e. 
_V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r ) ) ) `
 n ) )
231, 22wceq 1623 1  wff  ^ r  =  ( r  e. 
_V ,  n  e. 
NN0  |->  (  seq  0
( ( x  e. 
_V ,  y  e. 
_V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r ) ) ) `
 n ) )
Colors of variables: wff set class
This definition is referenced by:  relexp0  24025  relexpsucr  24026
  Copyright terms: Public domain W3C validator