Theorem relbrtpos

Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015)

	Ref	Expression
Assertion	relbrtpos	|- ( Rel F -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )

Proof

Step	Hyp	Ref	Expression
1		reltpos	|- Rel tpos F
2	1	a1i	|- ( Rel F -> Rel tpos F )
3		brrelex2	|- ( ( Rel tpos F /\ <. A , B >. tpos F C ) -> C e. _V )
4	2 3	sylan	|- ( ( Rel F /\ <. A , B >. tpos F C ) -> C e. _V )
5		brrelex2	|- ( ( Rel F /\ <. B , A >. F C ) -> C e. _V )
6		brtpos	|- ( C e. _V -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )
7	4 5 6	pm5.21nd	|- ( Rel F -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) )