Definition df-iota 6449

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 6467); otherwise, it evaluates to the empty set (see iotanul 6473). 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 7333 (or iotacl 6479 for unbounded iota), as demonstrated in the proof of supub 9366. This can be easier than applying riotasbc 7335 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 6447	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2715	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1541	. . . . . 6 class 𝑦
7	6	csn 4581	. . . . 5 class {𝑦}
8	4, 7	wceq 1542	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2715	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4864	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1542	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6450  cbviotavw  6457  iotaeq  6461  iotabi  6462  iotaval2  6464  iotanul2  6466  dffv4  6832  dfiota3  36117  cbviotadavw  36465  sn-iotalemcor  42546  reuabaiotaiota  47400