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 ) , · ⟩ }
	catcofval.x	⊢ · ∈ V
Assertion	catcofval	⊢ · = ( comp ‘ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		catbas.c	⊢ 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
2		catcofval.x	⊢ · ∈ V
3		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
4	1 3	eqbrtri	⊢ 𝐶 Struct ⟨ 1 , ; 1 5 ⟩
5		ccoid	⊢ comp = Slot ( comp ‘ ndx )
6		snsstp3	⊢ { ⟨ ( comp ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
7	6 1	sseqtrri	⊢ { ⟨ ( comp ‘ ndx ) , · ⟩ } ⊆ 𝐶
8	4 5 7	strfv	⊢ ( · ∈ V → · = ( comp ‘ 𝐶 ) )
9	2 8	ax-mp	⊢ · = ( comp ‘ 𝐶 )