Definition df-iota 6479

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 6497); otherwise, it evaluates to the empty set (see iotanul 6503). 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 7371 (or iotacl 6509 for unbounded iota), as demonstrated in the proof of supub 9407. This can be easier than applying riotasbc 7373 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 6477	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2742	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1561	. . . . . 6 class 𝑦
7	6	csn 4584	. . . . 5 class {𝑦}
8	4, 7	wceq 1562	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2742	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4867	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1562	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6480  cbviotavw  6487  iotaeq  6491  iotabi  6492  iotaval2  6494  iotanul2  6496  dffv4  6866  dfiota3  36276  cbviotadavw  36634  sn-iotalemcor  42846  reuabaiotaiota  47686