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Definition df-iota 5410
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 5421); otherwise, it evaluates to the empty set (see iotanul 5425). 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 6555 (or iotacl 5433 for unbounded iota), as demonstrated in the proof of supub 7456. This can be easier than applying riotasbc 6557 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 5408 . 2  class  ( iota
x ph )
41, 2cab 2421 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1651 . . . . . 6  class  y
76csn 3806 . . . . 5  class  { y }
84, 7wceq 1652 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2421 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 4007 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1652 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5411  iotaeq  5418  iotabi  5419  dffv4  5717  dfiota3  25760
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