Definition df-iota 6525

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 6544); otherwise, it evaluates to the empty set (see iotanul 6551). 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 7421 (or iotacl 6559 for unbounded iota), as demonstrated in the proof of supub 9528. This can be easier than applying riotasbc 7423 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 6523	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2717	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1536	. . . . . 6 class 𝑦
7	6	csn 4648	. . . . 5 class {𝑦}
8	4, 7	wceq 1537	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2717	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4931	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1537	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6526  cbviotavw  6533  iotaeq  6538  iotabi  6539  iotaval2  6541  iotanul2  6543  dffv4  6917  dfiota3  35887  cbviotadavw  36235  sn-iotalemcor  42215  reuabaiotaiota  47002