Description: Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015) (Revised by Mario Carneiro, 21-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cnindis | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri | |
|
2 | topontop | |
|
3 | 2 | ad2antrr | |
4 | 0opn | |
|
5 | 3 4 | syl | |
6 | imaeq2 | |
|
7 | ima0 | |
|
8 | 6 7 | eqtrdi | |
9 | 8 | eleq1d | |
10 | 5 9 | syl5ibrcom | |
11 | fimacnv | |
|
12 | 11 | adantl | |
13 | toponmax | |
|
14 | 13 | ad2antrr | |
15 | 12 14 | eqeltrd | |
16 | imaeq2 | |
|
17 | 16 | eleq1d | |
18 | 15 17 | syl5ibrcom | |
19 | 10 18 | jaod | |
20 | 1 19 | syl5 | |
21 | 20 | ralrimiv | |
22 | 21 | ex | |
23 | 22 | pm4.71d | |
24 | id | |
|
25 | elmapg | |
|
26 | 24 13 25 | syl2anr | |
27 | indistopon | |
|
28 | iscn | |
|
29 | 27 28 | sylan2 | |
30 | 23 26 29 | 3bitr4rd | |
31 | 30 | eqrdv | |