Definition df-iota 5851

Description: Define Russell's definition description binder, which can be read as "the unique x such that ph," where ph ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that ph is true (see iotaval 5862); otherwise, it evaluates to the empty set (see iotanul 5866). Russell used the inverted iota symbol iota 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 6624 (or iotacl 5874 for unbounded iota), as demonstrated in the proof of supub 8365. This can be easier than applying riotasbc 6626 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	|- ( iota x ph ) = U. { y | { x | ph } = { y } }

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Detailed syntax breakdown of Definition df-iota

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3	1, 2	cio 5849	. 2 class ( iota x ph )
4	1, 2	cab 2608	. . . . 5 class { x | ph }
5		vy	. . . . . . 7 setvar y
6	5	cv 1482	. . . . . 6 class y
7	6	csn 4177	. . . . 5 class { y }
8	4, 7	wceq 1483	. . . 4 wff { x | ph } = { y }
9	8, 5	cab 2608	. . 3 class { y | { x | ph } = { y } }
10	9	cuni 4436	. 2 class U. { y | { x | ph } = { y } }
11	3, 10	wceq 1483	1 wff ( iota x ph ) = U. { y | { x | ph } = { y } }

This definition is referenced by:  dfiota2  5852  iotaeq  5859  iotabi  5860  dffv4  6188  dfiota3  32030