Definition df-iota 6515

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 6533); otherwise, it evaluates to the empty set (see iotanul 6540). 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 7403 (or iotacl 6548 for unbounded iota), as demonstrated in the proof of supub 9496. This can be easier than applying riotasbc 7405 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 6513	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2711	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1535	. . . . . 6 class 𝑦
7	6	csn 4630	. . . . 5 class {𝑦}
8	4, 7	wceq 1536	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2711	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4911	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1536	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6516  cbviotavw  6523  iotaeq  6527  iotabi  6528  iotaval2  6530  iotanul2  6532  dffv4  6903  dfiota3  35904  cbviotadavw  36251  sn-iotalemcor  42239  reuabaiotaiota  47036