Definition df-iota 6314

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 6329); otherwise, it evaluates to the empty set (see iotanul 6333). 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 7130 (or iotacl 6341 for unbounded iota), as demonstrated in the proof of supub 8923. This can be easier than applying riotasbc 7132 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 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Definition df-iota

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	cio 6312	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2799	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1536	. . . . . 6 class 𝑦
7	6	csn 4567	. . . . 5 class {𝑦}
8	4, 7	wceq 1537	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2799	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4838	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1537	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6315  iotaeq  6326  iotabi  6327  dffv4  6667  dfiota3  33384  reuabaiotaiota  43307