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Definition df-iota 5381
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 5392); otherwise, it evaluates to the empty set (see iotanul 5396). 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 6526 (or iotacl 5404 for unbounded iota), as demonstrated in the proof of supub 7424. This can be easier than applying riotasbc 6528 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 5379 . 2  class  ( iota
x ph )
41, 2cab 2394 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1648 . . . . . 6  class  y
76csn 3778 . . . . 5  class  { y }
84, 7wceq 1649 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2394 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3979 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1649 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5382  iotaeq  5389  iotabi  5390  dffv4  5688  dfiota3  25680
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