This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )