Definition df-iota 6376

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 6392); otherwise, it evaluates to the empty set (see iotanul 6396). 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 7229 (or iotacl 6404 for unbounded iota), as demonstrated in the proof of supub 9148. This can be easier than applying riotasbc 7231 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 6374	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2715	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1538	. . . . . 6 class 𝑦
7	6	csn 4558	. . . . 5 class {𝑦}
8	4, 7	wceq 1539	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2715	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4836	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1539	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6377  cbviotavw  6384  iotaeq  6389  iotabi  6390  dffv4  6753  dfiota3  34152  sn-iotalemcor  40118  sn-iotaval  40119  sn-iotanul  40121  reuabaiotaiota  44466