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Definition df-iota 5421
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 5432); otherwise, it evaluates to the empty set (see iotanul 5436). 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 riotacl2 6566 (or iotacl 5444 for unbounded iota), as demonstrated in the proof of supub 7467. This can be easier than applying riotasbc 6568 or 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  set  x
31, 2cio 5419 . 2  class  ( iota
x ph )
41, 2cab 2424 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1652 . . . . . 6  class  y
76csn 3816 . . . . 5  class  { y }
84, 7wceq 1653 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2424 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 4017 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1653 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5422  iotaeq  5429  iotabi  5430  dffv4  5728  dfiota3  25773
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