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Definition df-iota 6425
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 6444); otherwise, it evaluates to the empty set (see iotanul 6451). 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 7303 (or iotacl 6459 for unbounded iota), as demonstrated in the proof of supub 9308. This can be easier than applying riotasbc 7305 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 6423 . 2 class (℩𝑥𝜑)
41, 2cab 2713 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1539 . . . . . 6 class 𝑦
76csn 4572 . . . . 5 class {𝑦}
84, 7wceq 1540 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2713 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cuni 4851 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1540 1 wff (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  6426  cbviotavw  6433  iotaeq  6438  iotabi  6439  iotaval2  6441  iotanul2  6443  dffv4  6816  dfiota3  34316  sn-iotalemcor  40441  reuabaiotaiota  44919
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