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Definition df-iota 5768
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 5779); otherwise, it evaluates to the empty set (see iotanul 5783). 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 6524 (or iotacl 5791 for unbounded iota), as demonstrated in the proof of supub 8248. This can be easier than applying riotasbc 6526 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 5766 . 2 class (℩𝑥𝜑)
41, 2cab 2596 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1474 . . . . . 6 class 𝑦
76csn 4125 . . . . 5 class {𝑦}
84, 7wceq 1475 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2596 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4372 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1475 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  5769  iotaeq  5776  iotabi  5777  dffv4  6100  dfiota3  31200
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