# Metamath Proof Explorer

## Definition df-iota

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 ); otherwise, it evaluates to the empty set (see iotanul ). 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 (or iotacl for unbounded iota), as demonstrated in the proof of supub . This can be easier than applying riotasbc 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)

Ref Expression
Assertion df-iota
`|- ( iota x ph ) = U. { y | { x | ph } = { y } }`

### Detailed syntax breakdown

Step Hyp Ref Expression
0 vx
` |-  x`
1 wph
` |-  ph`
2 1 0 cio
` |-  ( iota x ph )`
3 vy
` |-  y`
4 1 0 cab
` |-  { x | ph }`
5 3 cv
` |-  y`
6 5 csn
` |-  { y }`
7 4 6 wceq
` |-  { x | ph } = { y }`
8 7 3 cab
` |-  { y | { x | ph } = { y } }`
9 8 cuni
` |-  U. { y | { x | ph } = { y } }`
10 2 9 wceq
` |-  ( iota x ph ) = U. { y | { x | ph } = { y } }`