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Definition df-iota 6004
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 6015); otherwise, it evaluates to the empty set (see iotanul 6019). 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 6779 (or iotacl 6027 for unbounded iota), as demonstrated in the proof of supub 8522. This can be easier than applying riotasbc 6781 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 6002 . 2 class (℩𝑥𝜑)
41, 2cab 2738 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1623 . . . . . 6 class 𝑦
76csn 4313 . . . . 5 class {𝑦}
84, 7wceq 1624 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2738 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4580 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1624 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  6005  iotaeq  6012  iotabi  6013  dffv4  6341  dfiota3  32328
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