Metamath Proof Explorer


Theorem homarel

Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypothesis homahom.h H=HomaC
Assertion homarel RelXHY

Proof

Step Hyp Ref Expression
1 homahom.h H=HomaC
2 xpss BaseC×BaseC×VV×V
3 eqid BaseC=BaseC
4 1 homarcl fXHYCCat
5 1 3 4 homaf fXHYH:BaseC×BaseC𝒫BaseC×BaseC×V
6 1 3 homarcl2 fXHYXBaseCYBaseC
7 6 simpld fXHYXBaseC
8 6 simprd fXHYYBaseC
9 5 7 8 fovcdmd fXHYXHY𝒫BaseC×BaseC×V
10 elelpwi fXHYXHY𝒫BaseC×BaseC×VfBaseC×BaseC×V
11 9 10 mpdan fXHYfBaseC×BaseC×V
12 2 11 sselid fXHYfV×V
13 12 ssriv XHYV×V
14 df-rel RelXHYXHYV×V
15 13 14 mpbir RelXHY