Definition df-iota 6446

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 6464); otherwise, it evaluates to the empty set (see iotanul 6470). 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 7329 (or iotacl 6476 for unbounded iota), as demonstrated in the proof of supub 9360. This can be easier than applying riotasbc 7331 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 6444	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2712	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1540	. . . . . 6 class 𝑦
7	6	csn 4578	. . . . 5 class {𝑦}
8	4, 7	wceq 1541	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2712	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4861	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1541	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6447  cbviotavw  6454  iotaeq  6458  iotabi  6459  iotaval2  6461  iotanul2  6463  dffv4  6829  dfiota3  36064  cbviotadavw  36412  sn-iotalemcor  42420  reuabaiotaiota  47275