Metamath Proof Explorer


Theorem idacd

Description: Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses idafval.i I = Id a C
idafval.b B = Base C
idafval.c φ C Cat
idahom.x φ X B
Assertion idacd φ cod a I X = X

Proof

Step Hyp Ref Expression
1 idafval.i I = Id a C
2 idafval.b B = Base C
3 idafval.c φ C Cat
4 idahom.x φ X B
5 eqid Hom a C = Hom a C
6 1 2 3 4 5 idahom φ I X X Hom a C X
7 5 homacd I X X Hom a C X cod a I X = X
8 6 7 syl φ cod a I X = X