Definition df-iota 6509

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 6527); otherwise, it evaluates to the empty set (see iotanul 6534). 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 7413 (or iotacl 6540 for unbounded iota), as demonstrated in the proof of supub 9475. This can be easier than applying riotasbc 7415 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 6507	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2741	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1569	. . . . . 6 class 𝑦
7	6	csn 4588	. . . . 5 class {𝑦}
8	4, 7	wceq 1570	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2741	. . 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  6510  cbviotavw  6517  iotaeq  6521  iotabi  6522  iotaval2  6524  iotanul2  6526  dffv4  6902  dfiota3  36862  cbviotadavw  37235  sn-iotalemcor  43467  reuabaiotaiota  48360