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Definition df-iota 4810
Description: Define Russell's definition description binder, which can be read as "the unique x such that φ," where φ ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that φ is true (see iotaval 4821); otherwise, it evaluates to the empty set (see iotanul 4825). Russell used the inverted iota symbol 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 4833 (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 (℩xφ) = {y ∣ {xφ} = {y}}
Distinct variable groups:   x,y   φ,y
Allowed substitution hint:   φ(x)

Detailed syntax breakdown of Definition df-iota
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 setvar x
31, 2cio 4808 . 2 class (℩xφ)
41, 2cab 2023 . . . . 5 class {xφ}
5 vy . . . . . . 7 setvar y
65cv 1241 . . . . . 6 class y
76csn 3367 . . . . 5 class {y}
84, 7wceq 1242 . . . 4 wff {xφ} = {y}
98, 5cab 2023 . . 3 class {y ∣ {xφ} = {y}}
109cuni 3571 . 2 class {y ∣ {xφ} = {y}}
113, 10wceq 1242 1 wff (℩xφ) = {y ∣ {xφ} = {y}}
Colors of variables: wff set class
This definition is referenced by:  dfiota2  4811  iotaeq  4818  iotabi  4819  iotass  4827  dffv4g  5118  nfvres  5149
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