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Metamath Proof Explorer

Theorem relbrcnvg

Description: When R is a relation, the sethood assumptions on brcnv can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion relbrcnvg 
|- ( Rel R -> ( A `' R B <-> B R A ) )

Proof

Step Hyp Ref Expression
1 relcnv 
 |-  Rel `' R
2 1 brrelex12i 
 |-  ( A `' R B -> ( A e. _V /\ B e. _V ) )
3 2 a1i 
 |-  ( Rel R -> ( A `' R B -> ( A e. _V /\ B e. _V ) ) )
4 brrelex12 
 |-  ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
5 4 ancomd 
 |-  ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
6 5 ex 
 |-  ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
7 brcnvg 
 |-  ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) )
8 7 a1i 
 |-  ( Rel R -> ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) ) )
9 3 6 8 pm5.21ndd 
 |-  ( Rel R -> ( A `' R B <-> B R A ) )