Definition df-iota 6514

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 6532); otherwise, it evaluates to the empty set (see iotanul 6539). 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 7404 (or iotacl 6547 for unbounded iota), as demonstrated in the proof of supub 9499. This can be easier than applying riotasbc 7406 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 6512	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2714	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1539	. . . . . 6 class 𝑦
7	6	csn 4626	. . . . 5 class {𝑦}
8	4, 7	wceq 1540	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2714	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4907	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1540	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6515  cbviotavw  6522  iotaeq  6526  iotabi  6527  iotaval2  6529  iotanul2  6531  dffv4  6903  dfiota3  35924  cbviotadavw  36270  sn-iotalemcor  42261  reuabaiotaiota  47099