Definition df-iota 6437

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 6455); otherwise, it evaluates to the empty set (see iotanul 6461). 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 7319 (or iotacl 6467 for unbounded iota), as demonstrated in the proof of supub 9343. This can be easier than applying riotasbc 7321 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 6435	. 2 class (℩𝑥𝜑)
4	1, 2	cab 2709	. . . . 5 class {𝑥 ∣ 𝜑}
5		vy	. . . . . . 7 setvar 𝑦
6	5	cv 1540	. . . . . 6 class 𝑦
7	6	csn 4573	. . . . 5 class {𝑦}
8	4, 7	wceq 1541	. . . 4 wff {𝑥 ∣ 𝜑} = {𝑦}
9	8, 5	cab 2709	. . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
10	9	cuni 4856	. 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
11	3, 10	wceq 1541	1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}

This definition is referenced by:  dfiota2  6438  cbviotavw  6445  iotaeq  6449  iotabi  6450  iotaval2  6452  iotanul2  6454  dffv4  6819  dfiota3  35965  cbviotadavw  36311  sn-iotalemcor  42263  reuabaiotaiota  47126