Definition df-iota 6464

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 6482); otherwise, it evaluates to the empty set (see iotanul 6489). 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 7360 (or iotacl 6497 for unbounded iota), as demonstrated in the proof of supub 9410. This can be easier than applying riotasbc 7362 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 6462	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2707	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1539	. . . . . 6 class 𝑦
7	6	csn 4589	. . . . 5 class {𝑦}
8	4, 7	wceq 1540	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2707	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4871	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1540	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6465  cbviotavw  6472  iotaeq  6476  iotabi  6477  iotaval2  6479  iotanul2  6481  dffv4  6855  dfiota3  35911  cbviotadavw  36257  sn-iotalemcor  42210  reuabaiotaiota  47085