Definition df-iota 6454

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 6472); otherwise, it evaluates to the empty set (see iotanul 6478). 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 7340 (or iotacl 6484 for unbounded iota), as demonstrated in the proof of supub 9372. This can be easier than applying riotasbc 7342 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 6452	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2714	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1541	. . . . . 6 class 𝑦
7	6	csn 4567	. . . . 5 class {𝑦}
8	4, 7	wceq 1542	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2714	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4850	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1542	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6455  cbviotavw  6462  iotaeq  6466  iotabi  6467  iotaval2  6469  iotanul2  6471  dffv4  6837  dfiota3  36103  cbviotadavw  36451  sn-iotalemcor  42663  reuabaiotaiota  47535