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Definition df-iota 5091
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 5102); otherwise, it evaluates to the empty set (see iotanul 5106). 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 5114 (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 5089 . 2 class (℩𝑥𝜑)
41, 2cab 2125 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1330 . . . . . 6 class 𝑦
76csn 3527 . . . . 5 class {𝑦}
84, 7wceq 1331 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2125 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 3739 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1331 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5092  iotaeq  5099  iotabi  5100  iotass  5108  dffv4g  5421  nfvres  5457
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