Definition df-iota 6494

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 6512); otherwise, it evaluates to the empty set (see iotanul 6518). 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 7393 (or iotacl 6524 for unbounded iota), as demonstrated in the proof of supub 9451. This can be easier than applying riotasbc 7395 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 6492	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2738	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1568	. . . . . 6 class 𝑦
7	6	csn 4583	. . . . 5 class {𝑦}
8	4, 7	wceq 1569	. . . 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 1569	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is used by:  dfiota2  6495  cbviotavw  6502  iotaeq  6506  iotabi  6507  iotaval2  6509  iotanul2  6511  dffv4  6882  dfiota3  36727  cbviotadavw  37100  sn-iotalemcor  43319  reuabaiotaiota  48199