Definition df-iota 6483

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 6501); otherwise, it evaluates to the empty set (see iotanul 6507). 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 7381 (or iotacl 6513 for unbounded iota), as demonstrated in the proof of supub 9429. This can be easier than applying riotasbc 7383 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 6481	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2738	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1569	. . . . . 6 class 𝑦
7	6	csn 4583	. . . . 5 class {𝑦}
8	4, 7	wceq 1570	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2738	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4866	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1570	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is used by:  dfiota2  6484  cbviotavw  6491  iotaeq  6495  iotabi  6496  iotaval2  6498  iotanul2  6500  dffv4  6870  dfiota3  36607  cbviotadavw  36980  sn-iotalemcor  43196  reuabaiotaiota  48079