Theorem catbas

Description: The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025)

	Ref	Expression
Hypotheses	catbas.c	|- C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. }
	catbas.b	|- B e. _V
Assertion	catbas	|- B = ( Base ` C )

Proof

Step	Hyp	Ref	Expression
1		catbas.c	|- C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. }
2		catbas.b	|- B e. _V
3		catstr	|- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. } Struct <. 1 , ; 1 5 >.
4	1 3	eqbrtri	|- C Struct <. 1 , ; 1 5 >.
5		baseid	|- Base = Slot ( Base ` ndx )
6		snsstp1	|- { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. }
7	6 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. } C_ C
8	4 5 7	strfv	|- ( B e. _V -> B = ( Base ` C ) )
9	2 8	ax-mp	|- B = ( Base ` C )