Theorem prstcbas

Description: The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024)

	Ref	Expression
Hypotheses	prstcnid.c	|- ( ph -> C = ( ProsetToCat ` K ) )
	prstcnid.k	|- ( ph -> K e. Proset )
	prstcbas.b	|- ( ph -> B = ( Base ` K ) )
Assertion	prstcbas	|- ( ph -> B = ( Base ` C ) )

Proof

Step	Hyp	Ref	Expression
1		prstcnid.c	|- ( ph -> C = ( ProsetToCat ` K ) )
2		prstcnid.k	|- ( ph -> K e. Proset )
3		prstcbas.b	|- ( ph -> B = ( Base ` K ) )
4		baseid	|- Base = Slot ( Base ` ndx )
5		slotsbhcdif	|- ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= ( comp ` ndx ) /\ ( Hom ` ndx ) =/= ( comp ` ndx ) )
6	5	simp2i	|- ( Base ` ndx ) =/= ( comp ` ndx )
7	5	simp1i	|- ( Base ` ndx ) =/= ( Hom ` ndx )
8	1 2 4 6 7	prstcnid	|- ( ph -> ( Base ` K ) = ( Base ` C ) )
9	3 8	eqtrd	|- ( ph -> B = ( Base ` C ) )