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Metamath Proof Explorer

Theorem brerser

Description: Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when A and R are sets. (Contributed by Peter Mazsa, 25-Aug-2021)

Ref Expression
Assertion brerser 
|- ( ( A e. V /\ R e. W ) -> ( R Ers A <-> R ErALTV A ) )

Proof

Step Hyp Ref Expression
1 brers 
 |-  ( A e. V -> ( R Ers A <-> ( R e. EqvRels /\ R DomainQss A ) ) )
2 1 adantr 
 |-  ( ( A e. V /\ R e. W ) -> ( R Ers A <-> ( R e. EqvRels /\ R DomainQss A ) ) )
3 eleqvrelsrel 
 |-  ( R e. W -> ( R e. EqvRels <-> EqvRel R ) )
4 3 adantl 
 |-  ( ( A e. V /\ R e. W ) -> ( R e. EqvRels <-> EqvRel R ) )
5 brdmqssqs 
 |-  ( ( A e. V /\ R e. W ) -> ( R DomainQss A <-> R DomainQs A ) )
6 4 5 anbi12d 
 |-  ( ( A e. V /\ R e. W ) -> ( ( R e. EqvRels /\ R DomainQss A ) <-> ( EqvRel R /\ R DomainQs A ) ) )
7 df-erALTV 
 |-  ( R ErALTV A <-> ( EqvRel R /\ R DomainQs A ) )
8 6 7 bitr4di 
 |-  ( ( A e. V /\ R e. W ) -> ( ( R e. EqvRels /\ R DomainQss A ) <-> R ErALTV A ) )
9 2 8 bitrd 
 |-  ( ( A e. V /\ R e. W ) -> ( R Ers A <-> R ErALTV A ) )