Theorem oppfrcl

Description: If an opposite functor of a class is a functor, then the original class must be an ordered pair. (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 )
Assertion	oppfrcl	|- ( ph -> F e. ( _V X. _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	1 2	oppfrcllem	|- ( ph -> G =/= (/) )
5		ndmfv	|- ( -. F e. dom oppFunc -> ( oppFunc ` F ) = (/) )
6	3 5	eqtrid	|- ( -. F e. dom oppFunc -> G = (/) )
7	6	necon1ai	|- ( G =/= (/) -> F e. dom oppFunc )
8	4 7	syl	|- ( ph -> F e. dom oppFunc )
9		df-oppf	|- oppFunc = ( f e. _V , g e. _V |-> if ( ( Rel g /\ Rel dom g ) , <. f , tpos g >. , (/) ) )
10		opex	|- <. f , tpos g >. e. _V
11		0ex	|- (/) e. _V
12	10 11	ifex	|- if ( ( Rel g /\ Rel dom g ) , <. f , tpos g >. , (/) ) e. _V
13	9 12	dmmpo	|- dom oppFunc = ( _V X. _V )
14	8 13	eleqtrdi	|- ( ph -> F e. ( _V X. _V ) )