Description: If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | clsnei.o | |
|
clsnei.p | |
||
clsnei.d | |
||
clsnei.f | |
||
clsnei.h | |
||
clsnei.r | |
||
Assertion | clsneikex | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.o | |
|
2 | clsnei.p | |
|
3 | clsnei.d | |
|
4 | clsnei.f | |
|
5 | clsnei.h | |
|
6 | clsnei.r | |
|
7 | 3 5 6 | clsneibex | |
8 | pwexg | |
|
9 | 8 | adantl | |
10 | simpr | |
|
11 | 1 9 10 4 | fsovf1od | |
12 | f1ofn | |
|
13 | 11 12 | syl | |
14 | 2 3 10 | dssmapf1od | |
15 | f1of | |
|
16 | 14 15 | syl | |
17 | 6 | adantr | |
18 | 5 | breqi | |
19 | 17 18 | sylib | |
20 | 13 16 19 | brcoffn | |
21 | 7 20 | mpdan | |
22 | 21 | simpld | |
23 | 2 3 22 | ntrclsiex | |