# Metamath Proof Explorer

## Definition df-aiota

Description: Alternate version of Russell's definition of a 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 aiotaval ); otherwise, it is not a set (see aiotaexb ), or even more concrete, it is the universe _V (see aiotavb ). Since this is an alternative for df-iota , we call this symbol iota' alternate iota in the following.

The advantage of this definition is the clear distinguishability of the defined and undefined cases: the alternate iota over a wff is defined iff it is a set (see aiotaexb ). With the original definition, there is no corresponding theorem ( E! x ph <-> ( iota x ph ) =/= (/) ) , because (/) can be a valid unique set satisfying a wff (see, for example, iota0def ). Only the right to left implication would hold, see (negated) iotanul . For defined cases, however, both definitions df-iota and df-aiota are equivalent, see reuaiotaiota . (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022)

Ref Expression
Assertion df-aiota ${⊢}\iota =\bigcap \left\{{y}|\left\{{x}|{\phi }\right\}=\left\{{y}\right\}\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 vx ${setvar}{x}$
1 wph ${wff}{\phi }$
2 1 0 caiota ${class}\iota$
3 vy ${setvar}{y}$
4 1 0 cab ${class}\left\{{x}|{\phi }\right\}$
5 3 cv ${setvar}{y}$
6 5 csn ${class}\left\{{y}\right\}$
7 4 6 wceq ${wff}\left\{{x}|{\phi }\right\}=\left\{{y}\right\}$
8 7 3 cab ${class}\left\{{y}|\left\{{x}|{\phi }\right\}=\left\{{y}\right\}\right\}$
9 8 cint ${class}\bigcap \left\{{y}|\left\{{x}|{\phi }\right\}=\left\{{y}\right\}\right\}$
10 2 9 wceq ${wff}\iota =\bigcap \left\{{y}|\left\{{x}|{\phi }\right\}=\left\{{y}\right\}\right\}$