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Definition df-iota 5235
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 5246); otherwise, it evaluates to the empty set (see iotanul 5250). 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 6334 (or iotacl 5258 for unbounded iota), as demonstrated in the proof of supub 7226. This can be easier than applying riotasbc 6336 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 5233 . 2  class  ( iota
x ph )
41, 2cab 2282 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1631 . . . . . 6  class  y
76csn 3653 . . . . 5  class  { y }
84, 7wceq 1632 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2282 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3843 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1632 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5236  iotaeq  5243  iotabi  5244  dffv4  5538  dfiota3  24532
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