Metamath Proof Explorer

Theorem symrelim

Description: Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021)

Ref Expression
Assertion symrelim 
|- ( SymRel R -> dom R = ran R )

Proof

Step Hyp Ref Expression
1 rncnv 
 |-  ran `' R = dom R
2 dfsymrel4 
 |-  ( SymRel R <-> ( `' R = R /\ Rel R ) )
3 2 simplbi 
 |-  ( SymRel R -> `' R = R )
4 3 rneqd 
 |-  ( SymRel R -> ran `' R = ran R )
5 1 4 eqtr3id 
 |-  ( SymRel R -> dom R = ran R )