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Definition df-iota 6254
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 6265); otherwise, it evaluates to the empty set (see iotanul 6269). 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 6315 (or iotacl 6277 for unbounded iota), as demonstrated in the proof of supub 7207. This can be easier than applying riotasbc 6317 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 group:    ph( x)

Detailed syntax breakdown of Definition df-iota
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
31, 2cio 6252 . 2  class  ( iota
x ph )
41, 2cab 2272 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1624 . . . . . 6  class  y
76csn 3643 . . . . 5  class  { y }
84, 7wceq 1625 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2272 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3830 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1625 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  6255  iotaeq  6262  iotabi  6263  dffv3  6282  dfiota3  23871
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