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

Definition df-riota 6258
Description: Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse  A. See also comments for df-iota 6211. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
df-riota  |-  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )

Detailed syntax breakdown of Definition df-riota
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
3 cA . . 3  class  A
41, 2, 3crio 6249 . 2  class  ( iota_ x  e.  A ph )
51, 2, 3wreu 2518 . . 3  wff  E! x  e.  A  ph
62cv 1618 . . . . . 6  class  x
76, 3wcel 1621 . . . . 5  wff  x  e.  A
87, 1wa 360 . . . 4  wff  ( x  e.  A  /\  ph )
98, 2cio 6209 . . 3  class  ( iota
x ( x  e.  A  /\  ph )
)
107, 2cab 2242 . . . 4  class  { x  |  x  e.  A }
11 cund 6248 . . . 4  class  Undef
1210, 11cfv 4659 . . 3  class  ( Undef `  { x  |  x  e.  A } )
135, 9, 12cif 3525 . 2  class  if ( E! x  e.  A  ph ,  ( iota x
( x  e.  A  /\  ph ) ) ,  ( Undef `  { x  |  x  e.  A } ) )
144, 13wceq 1619 1  wff  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
Colors of variables: wff set class
This definition is referenced by:  riotaeqdv  6259  riotabidv  6260  riotaex  6262  riotav  6263  riotaiota  6264  nfriota1  6266  nfriotad  6267  cbvriota  6269  riotabidva  6275  riotaund  6295
  Copyright terms: Public domain W3C validator