Theorem catcofval

Description: Composition 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. >. }
	catcofval.x	|- .x. e. _V
Assertion	catcofval	|- .x. = ( comp ` C )

Proof

Step	Hyp	Ref	Expression
1		catbas.c	|- C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. }
2		catcofval.x	|- .x. 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		ccoid	|- comp = Slot ( comp ` ndx )
6		snsstp3	|- { <. ( comp ` ndx ) , .x. >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. }
7	6 1	sseqtrri	|- { <. ( comp ` ndx ) , .x. >. } C_ C
8	4 5 7	strfv	|- ( .x. e. _V -> .x. = ( comp ` C ) )
9	2 8	ax-mp	|- .x. = ( comp ` C )