Theorem cathomfval

Description: The hom-sets 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 ) , · ⟩ }
	cathomfval.h	⊢ 𝐻 ∈ V
Assertion	cathomfval	⊢ 𝐻 = ( Hom ‘ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		catbas.c	⊢ 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
2		cathomfval.h	⊢ 𝐻 ∈ 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		homid	⊢ Hom = Slot ( Hom ‘ ndx )
6		snsstp2	⊢ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
7	6 1	sseqtrri	⊢ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ } ⊆ 𝐶
8	4 5 7	strfv	⊢ ( 𝐻 ∈ V → 𝐻 = ( Hom ‘ 𝐶 ) )
9	2 8	ax-mp	⊢ 𝐻 = ( Hom ‘ 𝐶 )