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 6508). 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 7376 (or iotacl 6516 for unbounded iota), as demonstrated in the proof of supub 9469. This can be easier than applying riotasbc 7378 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 2713	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1539	. . . . . 6 class 𝑦
7	6	csn 4601	. . . . 5 class {𝑦}
8	4, 7	wceq 1540	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2713	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4883	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1540	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6484  cbviotavw  6491  iotaeq  6495  iotabi  6496  iotaval2  6498  iotanul2  6500  dffv4  6872  dfiota3  35887  cbviotadavw  36233  sn-iotalemcor  42219  reuabaiotaiota  47064