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Definition df-iota 4894
Description: Define Russell's definition description binder, which can be read as "the unique  x such that  ph," where  ph ordinarily contains  x as a free variable. Our definition is meaningful only when there is exactly one  x such that  ph is true (see iotaval 4905); otherwise, it evaluates to the empty set (see iotanul 4909). Russell used the inverted iota symbol 
iota to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4917 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion
Ref Expression
df-iota  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Detailed syntax breakdown of Definition df-iota
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  setvar  x
31, 2cio 4892 . 2  class  ( iota
x ph )
41, 2cab 2042 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  setvar  y
65cv 1258 . . . . . 6  class  y
76csn 3402 . . . . 5  class  { y }
84, 7wceq 1259 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2042 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3607 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1259 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  4895  iotaeq  4902  iotabi  4903  iotass  4911  dffv4g  5202  nfvres  5233
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