Definition df-iota 6384

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 6400); otherwise, it evaluates to the empty set (see iotanul 6404). 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 7241 (or iotacl 6412 for unbounded iota), as demonstrated in the proof of supub 9205. This can be easier than applying riotasbc 7243 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 6382	. 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 4561	. . . . 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 4839	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1539	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6385  cbviotavw  6392  iotaeq  6397  iotabi  6398  dffv4  6763  dfiota3  34233  sn-iotalemcor  40198  iotavallem  40200  sn-iotanul  40202  reuabaiotaiota  44557