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Definition df-iota 5401
Description: Define Russell's definition description binder, which can be read as "the unique such that ," where ordinarily contains as a free variable. Our definition is meaningful only when there is exactly one such that is true (see iotaval 5412); otherwise, it evaluates to the empty set (see iotanul 5416). 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 6077 (or iotacl 5424 for unbounded iota), as demonstrated in the proof of supub 7631. This can be easier than applying riotasbc 6079 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
Distinct variable groups:   ,   ,
Allowed substitution hint:   ( )

Detailed syntax breakdown of Definition df-iota
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2cio 5399 . 2
41, 2cab 2475 . . . . 5
5 vy . . . . . . 7
65cv 1669 . . . . . 6
76csn 3909 . . . . 5
84, 7wceq 1670 . . . 4
98, 5cab 2475 . . 3
109cuni 4117 . 2
113, 10wceq 1670 1
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5402  iotaeq  5409  iotabi  5410  dffv4  5705  dfiota3  27107
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