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Definition df-iota 5753
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 5764); otherwise, it evaluates to the empty set (see iotanul 5768). 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 6501 (or iotacl 5776 for unbounded iota), as demonstrated in the proof of supub 8225. This can be easier than applying riotasbc 6503 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 5751 . 2 class (℩𝑥𝜑)
41, 2cab 2595 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1473 . . . . . 6 class 𝑦
76csn 4124 . . . . 5 class {𝑦}
84, 7wceq 1474 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2595 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4366 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1474 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  5754  iotaeq  5761  iotabi  5762  dffv4  6084  dfiota3  30993
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