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Definition df-trcl 13052
Description: Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
Assertion
Ref Expression
df-trcl  |-  t+  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  ( z  o.  z
)  C_  z ) } )
Distinct variable group:    x, z

Detailed syntax breakdown of Definition df-trcl
StepHypRef Expression
1 ctcl 13050 . 2  class  t+
2 vx . . 3  setvar  x
3 cvv 3080 . . 3  class  _V
42cv 1436 . . . . . . 7  class  x
5 vz . . . . . . . 8  setvar  z
65cv 1436 . . . . . . 7  class  z
74, 6wss 3436 . . . . . 6  wff  x  C_  z
86, 6ccom 4857 . . . . . . 7  class  ( z  o.  z )
98, 6wss 3436 . . . . . 6  wff  ( z  o.  z )  C_  z
107, 9wa 370 . . . . 5  wff  ( x 
C_  z  /\  (
z  o.  z ) 
C_  z )
1110, 5cab 2407 . . . 4  class  { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) }
1211cint 4255 . . 3  class  |^| { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) }
132, 3, 12cmpt 4482 . 2  class  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) } )
141, 13wceq 1437 1  wff  t+  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  ( z  o.  z
)  C_  z ) } )
Colors of variables: wff setvar class
This definition is referenced by:  trclfv  13065  dftrcl3  36283
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