Definition df-iota 6443

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 6461); otherwise, it evaluates to the empty set (see iotanul 6467). 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 7329 (or iotacl 6473 for unbounded iota), as demonstrated in the proof of supub 9361. This can be easier than applying riotasbc 7331 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 6441	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2713	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1541	. . . . . 6 class 𝑦
7	6	csn 4557	. . . . 5 class {𝑦}
8	4, 7	wceq 1542	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2713	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4840	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1542	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6444  cbviotavw  6451  iotaeq  6455  iotabi  6456  iotaval2  6458  iotanul2  6460  dffv4  6826  dfiota3  36091  cbviotadavw  36439  sn-iotalemcor  42651  reuabaiotaiota  47523