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Definition df-iota 6308
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 6323); otherwise, it evaluates to the empty set (see iotanul 6327). 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 7119 (or iotacl 6335 for unbounded iota), as demonstrated in the proof of supub 8912. This can be easier than applying riotasbc 7121 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 6306 . 2 class (℩𝑥𝜑)
41, 2cab 2799 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1527 . . . . . 6 class 𝑦
76csn 4559 . . . . 5 class {𝑦}
84, 7wceq 1528 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2799 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4832 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1528 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  6309  iotaeq  6320  iotabi  6321  dffv4  6661  dfiota3  33282  reuabaiotaiota  43168
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