# Metamath Proof Explorer

## Theorem ecdmn0

Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996) (Revised by Mario Carneiro, 9-Jul-2014)

Ref Expression
Assertion ecdmn0
`|- ( A e. dom R <-> [ A ] R =/= (/) )`

### Proof

Step Hyp Ref Expression
1 elex
` |-  ( A e. dom R -> A e. _V )`
2 n0
` |-  ( [ A ] R =/= (/) <-> E. x x e. [ A ] R )`
3 ecexr
` |-  ( x e. [ A ] R -> A e. _V )`
4 3 exlimiv
` |-  ( E. x x e. [ A ] R -> A e. _V )`
5 2 4 sylbi
` |-  ( [ A ] R =/= (/) -> A e. _V )`
6 vex
` |-  x e. _V`
7 elecg
` |-  ( ( x e. _V /\ A e. _V ) -> ( x e. [ A ] R <-> A R x ) )`
8 6 7 mpan
` |-  ( A e. _V -> ( x e. [ A ] R <-> A R x ) )`
9 8 exbidv
` |-  ( A e. _V -> ( E. x x e. [ A ] R <-> E. x A R x ) )`
10 2 a1i
` |-  ( A e. _V -> ( [ A ] R =/= (/) <-> E. x x e. [ A ] R ) )`
11 eldmg
` |-  ( A e. _V -> ( A e. dom R <-> E. x A R x ) )`
12 9 10 11 3bitr4rd
` |-  ( A e. _V -> ( A e. dom R <-> [ A ] R =/= (/) ) )`
13 1 5 12 pm5.21nii
` |-  ( A e. dom R <-> [ A ] R =/= (/) )`