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Definition df-iota 5820
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 5831); otherwise, it evaluates to the empty set (see iotanul 5835). 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 riotacl2 6589 (or iotacl 5843 for unbounded iota), as demonstrated in the proof of supub 8325. This can be easier than applying riotasbc 6591 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 (℩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 5818 . 2 class (℩𝑥𝜑)
41, 2cab 2607 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1479 . . . . . 6 class 𝑦
76csn 4155 . . . . 5 class {𝑦}
84, 7wceq 1480 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2607 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4409 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1480 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  5821  iotaeq  5828  iotabi  5829  dffv4  6155  dfiota3  31725
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