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Metamath Proof Explorer

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	$⊢ φ \to G \in R$
		oppfrcl.2	$⊢ Rel R$
		oppfrcl.3	No typesetting found for \|- G = ( oppFunc ` F ) with typecode \|-
	Assertion	oppfrcl	$⊢ φ \to F \in (V \times V)$

Proof

Step	Hyp	Ref	Expression
1		oppfrcl.1	$⊢ φ \to G \in R$
2		oppfrcl.2	$⊢ Rel R$
3		oppfrcl.3	Could not format G = ( oppFunc ` F ) : No typesetting found for \|- G = ( oppFunc ` F ) with typecode \|-
4	1 2	oppfrcllem	$⊢ φ \to G \neq \emptyset$
5		ndmfv	Could not format ( -. F e. dom oppFunc -> ( oppFunc ` F ) = (/) ) : No typesetting found for \|- ( -. F e. dom oppFunc -> ( oppFunc ` F ) = (/) ) with typecode \|-
6	3 5	eqtrid	Could not format ( -. F e. dom oppFunc -> G = (/) ) : No typesetting found for \|- ( -. F e. dom oppFunc -> G = (/) ) with typecode \|-
7	6	necon1ai	Could not format ( G =/= (/) -> F e. dom oppFunc ) : No typesetting found for \|- ( G =/= (/) -> F e. dom oppFunc ) with typecode \|-
8	4 7	syl	Could not format ( ph -> F e. dom oppFunc ) : No typesetting found for \|- ( ph -> F e. dom oppFunc ) with typecode \|-
9		oppffn	Could not format oppFunc Fn ( _V X. _V ) : No typesetting found for \|- oppFunc Fn ( _V X. _V ) with typecode \|-
10	9	fndmi	Could not format dom oppFunc = ( _V X. _V ) : No typesetting found for \|- dom oppFunc = ( _V X. _V ) with typecode \|-
11	8 10	eleqtrdi	$⊢ φ \to F \in (V \times V)$