ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-iota Unicode version

Definition df-iota 4845
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 4856); otherwise, it evaluates to the empty set (see iotanul 4860). 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 iotacl 4868 (for unbounded iota). This can be easier than applying 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  setvar  x
31, 2cio 4843 . 2  class  ( iota
x ph )
41, 2cab 2026 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  setvar  y
65cv 1242 . . . . . 6  class  y
76csn 3372 . . . . 5  class  { y }
84, 7wceq 1243 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2026 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3577 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1243 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  4846  iotaeq  4853  iotabi  4854  iotass  4862  dffv4g  5153  nfvres  5184
  Copyright terms: Public domain W3C validator