Metamath Proof Explorer


Theorem prstchom

Description: Hom-sets of the constructed category are dependent on the preorder.

Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat . However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024)

Ref Expression
Hypotheses prstcnid.c No typesetting found for |- ( ph -> C = ( ProsetToCat ` K ) ) with typecode |-
prstcnid.k φKProset
prstchom.l φ˙=C
prstchom.e φH=HomC
prstchom.x φXBaseC
prstchom.y φYBaseC
Assertion prstchom φX˙YXHY

Proof

Step Hyp Ref Expression
1 prstcnid.c Could not format ( ph -> C = ( ProsetToCat ` K ) ) : No typesetting found for |- ( ph -> C = ( ProsetToCat ` K ) ) with typecode |-
2 prstcnid.k φKProset
3 prstchom.l φ˙=C
4 prstchom.e φH=HomC
5 prstchom.x φXBaseC
6 prstchom.y φYBaseC
7 1 2 3 prstchomval φ˙×1𝑜=HomC
8 4 7 eqtr4d φH=˙×1𝑜
9 1oex 1𝑜V
10 9 a1i φ1𝑜V
11 1n0 1𝑜
12 11 a1i φ1𝑜
13 8 10 12 fvconstrn0 φX˙YXHY