Definition df-iota 6442

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 6460); otherwise, it evaluates to the empty set (see iotanul 6466). 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 7326 (or iotacl 6472 for unbounded iota), as demonstrated in the proof of supub 9368. This can be easier than applying riotasbc 7328 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 6440	. 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 4579	. . . . 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 4861	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1540	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6443  cbviotavw  6450  iotaeq  6454  iotabi  6455  iotaval2  6457  iotanul2  6459  dffv4  6823  dfiota3  35899  cbviotadavw  36245  sn-iotalemcor  42198  reuabaiotaiota  47075