Definition df-iota 6492

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 6510); otherwise, it evaluates to the empty set (see iotanul 6516). 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 7383 (or iotacl 6522 for unbounded iota), as demonstrated in the proof of supub 9418. This can be easier than applying riotasbc 7385 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 6490	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2739	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1567	. . . . . 6 class 𝑦
7	6	csn 4588	. . . . 5 class {𝑦}
8	4, 7	wceq 1568	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2739	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4871	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1568	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6493  cbviotavw  6500  iotaeq  6504  iotabi  6505  iotaval2  6507  iotanul2  6509  dffv4  6878  dfiota3  36367  cbviotadavw  36725  sn-iotalemcor  42939  reuabaiotaiota  47769