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Definition df-riota 6304
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 5219. (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 6297 . 2  class  ( iota_ x  e.  A ph )
51, 2, 3wreu 2545 . . 3  wff  E! x  e.  A  ph
62cv 1622 . . . . . 6  class  x
76, 3wcel 1684 . . . . 5  wff  x  e.  A
87, 1wa 358 . . . 4  wff  ( x  e.  A  /\  ph )
98, 2cio 5217 . . 3  class  ( iota
x ( x  e.  A  /\  ph )
)
107, 2cab 2269 . . . 4  class  { x  |  x  e.  A }
11 cund 6296 . . . 4  class  Undef
1210, 11cfv 5255 . . 3  class  ( Undef `  { x  |  x  e.  A } )
135, 9, 12cif 3565 . 2  class  if ( E! x  e.  A  ph ,  ( iota x
( x  e.  A  /\  ph ) ) ,  ( Undef `  { x  |  x  e.  A } ) )
144, 13wceq 1623 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  6305  riotabidv  6306  riotaex  6308  riotav  6309  riotaiota  6310  nfriota1  6312  nfriotad  6313  cbvriota  6315  riotabidva  6321  riotaund  6341
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