Description: The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypothesis | homahom.h | |
|
Assertion | homadm | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homahom.h | |
|
2 | df-doma | |
|
3 | 2 | fveq1i | |
4 | fo1st | |
|
5 | fof | |
|
6 | 4 5 | ax-mp | |
7 | elex | |
|
8 | fvco3 | |
|
9 | 6 7 8 | sylancr | |
10 | 3 9 | eqtrid | |
11 | 1 | homarel | |
12 | 1st2ndbr | |
|
13 | 11 12 | mpan | |
14 | 1 | homa1 | |
15 | 13 14 | syl | |
16 | 15 | fveq2d | |
17 | eqid | |
|
18 | 1 17 | homarcl2 | |
19 | op1stg | |
|
20 | 18 19 | syl | |
21 | 10 16 20 | 3eqtrd | |