Definition df-iota 6390

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 6406); otherwise, it evaluates to the empty set (see iotanul 6410). 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 7245 (or iotacl 6418 for unbounded iota), as demonstrated in the proof of supub 9196. This can be easier than applying riotasbc 7247 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 6388	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2717	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1541	. . . . . 6 class 𝑦
7	6	csn 4567	. . . . 5 class {𝑦}
8	4, 7	wceq 1542	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2717	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4845	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1542	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6391  cbviotavw  6398  iotaeq  6403  iotabi  6404  dffv4  6768  dfiota3  34221  sn-iotalemcor  40187  iotavallem  40189  sn-iotanul  40191  reuabaiotaiota  44547