Definition df-iota 6493

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 6511); otherwise, it evaluates to the empty set (see iotanul 6517). 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 7389 (or iotacl 6523 for unbounded iota), as demonstrated in the proof of supub 9432. This can be easier than applying riotasbc 7391 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 6491	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2740	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1569	. . . . . 6 class 𝑦
7	6	csn 4587	. . . . 5 class {𝑦}
8	4, 7	wceq 1570	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2740	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4870	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1570	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is used by:  dfiota2  6494  cbviotavw  6501  iotaeq  6505  iotabi  6506  iotaval2  6508  iotanul2  6510  dffv4  6879  dfiota3  36487  cbviotadavw  36876  sn-iotalemcor  43079  reuabaiotaiota  47962