MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-trcl Unicode version

Definition df-trcl 20867
Description: Transitive closure of a relation. Experimental. (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 20865 . 2  class  t +
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
42cv 1631 . . . . . . 7  class  x
5 vz . . . . . . . 8  set  z
65cv 1631 . . . . . . 7  class  z
74, 6wss 3165 . . . . . 6  wff  x  C_  z
86, 6ccom 4709 . . . . . . 7  class  ( z  o.  z )
98, 6wss 3165 . . . . . 6  wff  ( z  o.  z )  C_  z
107, 9wa 358 . . . . 5  wff  ( x 
C_  z  /\  (
z  o.  z ) 
C_  z )
1110, 5cab 2282 . . . 4  class  { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) }
1211cint 3878 . . 3  class  |^| { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) }
132, 3, 12cmpt 4093 . 2  class  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  ( z  o.  z )  C_  z
) } )
141, 13wceq 1632 1  wff  t +  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  ( z  o.  z
)  C_  z ) } )
Colors of variables: wff set class
  Copyright terms: Public domain W3C validator