Definition df-iota 6444

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 6462); otherwise, it evaluates to the empty set (see iotanul 6468). 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 7332 (or iotacl 6474 for unbounded iota), as demonstrated in the proof of supub 9366. This can be easier than applying riotasbc 7334 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 6442	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2719	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1547	. . . . . 6 class 𝑦
7	6	csn 4557	. . . . 5 class {𝑦}
8	4, 7	wceq 1548	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2719	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4840	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1548	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6445  cbviotavw  6452  iotaeq  6456  iotabi  6457  iotaval2  6459  iotanul2  6461  dffv4  6827  dfiota3  36162  cbviotadavw  36510  sn-iotalemcor  42722  reuabaiotaiota  47562