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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 =/= (/) )