MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-iota Unicode version

Definition df-iota 5219
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 5230); otherwise, it evaluates to the empty set (see iotanul 5234). 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 6318 (or iotacl 5242 for unbounded iota), as demonstrated in the proof of supub 7210. This can be easier than applying riotasbc 6320 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 5217 . 2  class  ( iota
x ph )
41, 2cab 2269 . . . . 5  class  { x  |  ph }
5 vy . . . . . . 7  set  y
65cv 1622 . . . . . 6  class  y
76csn 3640 . . . . 5  class  { y }
84, 7wceq 1623 . . . 4  wff  { x  |  ph }  =  {
y }
98, 5cab 2269 . . 3  class  { y  |  { x  | 
ph }  =  {
y } }
109cuni 3827 . 2  class  U. {
y  |  { x  |  ph }  =  {
y } }
113, 10wceq 1623 1  wff  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5220  iotaeq  5227  iotabi  5228  dffv4  5522  dfiota3  23873
  Copyright terms: Public domain W3C validator