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Definition df-iota 5464
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 5475); otherwise, it evaluates to the empty set (see iotanul 5479). 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 6613 (or iotacl 5487 for unbounded iota), as demonstrated in the proof of supub 7513. This can be easier than applying riotasbc 6615 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 5462 . 2
41, 2cab 2429 . . . . 5
5 vy . . . . . . 7
65cv 1653 . . . . . 6
76csn 3841 . . . . 5
84, 7wceq 1654 . . . 4
98, 5cab 2429 . . 3
109cuni 4043 . 2
113, 10wceq 1654 1
Colors of variables: wff set class
This definition is referenced by:  dfiota2  5465  iotaeq  5472  iotabi  5473  dffv4  5772  dfiota3  25872
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