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Definition df-iota 6287
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 6302); otherwise, it evaluates to the empty set (see iotanul 6306). 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 7113 (or iotacl 6314 for unbounded iota), as demonstrated in the proof of supub 8911. This can be easier than applying riotasbc 7115 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 6285 . 2 class (℩𝑥𝜑)
41, 2cab 2779 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1537 . . . . . 6 class 𝑦
76csn 4528 . . . . 5 class {𝑦}
84, 7wceq 1538 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2779 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4803 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1538 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  6288  iotaeq  6299  iotabi  6300  dffv4  6646  dfiota3  33492  reuabaiotaiota  43631
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