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Definition df-iota 5447
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 5458); otherwise, it evaluates to the empty set (see iotanul 5462). 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 6592 (or iotacl 5470 for unbounded iota), as demonstrated in the proof of supub 7493. This can be easier than applying riotasbc 6594 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 5445 . 2  class  ( iota
x ph )
41, 2cab 2428 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1652 . . . . . 6  class  y
76csn 3838 . . . . 5  class  { y }
84, 7wceq 1653 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2428 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 4039 . 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  5448  iotaeq  5455  iotabi  5456  dffv4  5754  dfiota3  25799
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