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Definition df-iota 4867
 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 4878); otherwise, it evaluates to the empty set (see iotanul 4882). 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 4890 (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 4865 . 2 class (℩𝑥𝜑)
41, 2cab 2026 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1242 . . . . . 6 class 𝑦
76csn 3375 . . . . 5 class {𝑦}
84, 7wceq 1243 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2026 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 3580 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1243 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
 Colors of variables: wff set class This definition is referenced by:  dfiota2  4868  iotaeq  4875  iotabi  4876  iotass  4884  dffv4g  5175  nfvres  5206
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