Theorem parhomidval

Description: On the diagonal, the parallel-arrow hom operation has value { J } . (Contributed by Zhi Wang, 16-Sep-2026)

	Ref	Expression
Hypothesis	parhomval.h	|- H = ( x e. B , y e. B |-> if ( x = y , { J } , if ( ( x = X /\ y = Y ) , S , (/) ) ) )
Assertion	parhomidval	|- ( U e. B -> ( U H U ) = { J } )

Proof

Step	Hyp	Ref	Expression
1		parhomval.h	|- H = ( x e. B , y e. B |-> if ( x = y , { J } , if ( ( x = X /\ y = Y ) , S , (/) ) ) )
2		eqtr3	|- ( ( x = U /\ y = U ) -> x = y )
3	2	iftrued	|- ( ( x = U /\ y = U ) -> if ( x = y , { J } , if ( ( x = X /\ y = Y ) , S , (/) ) ) = { J } )
4		snex	|- { J } e. _V
5	3 1 4	ovmpoa	|- ( ( U e. B /\ U e. B ) -> ( U H U ) = { J } )
6	5	anidms	|- ( U e. B -> ( U H U ) = { J } )