Theorem oppfrcl2

Description: If an opposite functor of a class is a functor, then the two components of the original class must be sets. (Contributed by Zhi Wang, 14-Nov-2025)

	Ref	Expression
Hypotheses	oppfrcl.1	|- ( ph -> G e. R )
	oppfrcl.2	|- Rel R
	oppfrcl.3	|- G = ( oppFunc ` F )
	oppfrcl2.4	|- ( ph -> F = <. A , B >. )
Assertion	oppfrcl2	|- ( ph -> ( A e. _V /\ B e. _V ) )

Proof

Step	Hyp	Ref	Expression
1		oppfrcl.1	|- ( ph -> G e. R )
2		oppfrcl.2	|- Rel R
3		oppfrcl.3	|- G = ( oppFunc ` F )
4		oppfrcl2.4	|- ( ph -> F = <. A , B >. )
5	1 2 3	oppfrcl	|- ( ph -> F e. ( _V X. _V ) )
6	4 5	eqeltrrd	|- ( ph -> <. A , B >. e. ( _V X. _V ) )
7		0nelxp	|- -. (/) e. ( _V X. _V )
8		nelne2	|- ( ( <. A , B >. e. ( _V X. _V ) /\ -. (/) e. ( _V X. _V ) ) -> <. A , B >. =/= (/) )
9	6 7 8	sylancl	|- ( ph -> <. A , B >. =/= (/) )
10		opprc	|- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
11	10	necon1ai	|- ( <. A , B >. =/= (/) -> ( A e. _V /\ B e. _V ) )
12	9 11	syl	|- ( ph -> ( A e. _V /\ B e. _V ) )