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Metamath Proof Explorer

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 )