Description: Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009) (Proof shortened by Mario Carneiro, 25-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cncls | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnf2 | |
|
2 | 1 | 3expia | |
3 | elpwi | |
|
4 | 3 | adantl | |
5 | toponuni | |
|
6 | 5 | ad2antrr | |
7 | 4 6 | sseqtrd | |
8 | eqid | |
|
9 | 8 | cnclsi | |
10 | 9 | expcom | |
11 | 7 10 | syl | |
12 | 11 | ralrimdva | |
13 | 2 12 | jcad | |
14 | toponmax | |
|
15 | 14 | ad3antrrr | |
16 | cnvimass | |
|
17 | fdm | |
|
18 | 17 | ad2antlr | |
19 | 16 18 | sseqtrid | |
20 | 15 19 | sselpwd | |
21 | fveq2 | |
|
22 | 21 | imaeq2d | |
23 | imaeq2 | |
|
24 | 23 | fveq2d | |
25 | 22 24 | sseq12d | |
26 | 25 | rspcv | |
27 | 20 26 | syl | |
28 | topontop | |
|
29 | 28 | ad3antlr | |
30 | elpwi | |
|
31 | 30 | adantl | |
32 | toponuni | |
|
33 | 32 | ad3antlr | |
34 | 31 33 | sseqtrd | |
35 | ffun | |
|
36 | 35 | ad2antlr | |
37 | funimacnv | |
|
38 | 36 37 | syl | |
39 | inss1 | |
|
40 | 38 39 | eqsstrdi | |
41 | eqid | |
|
42 | 41 | clsss | |
43 | 29 34 40 42 | syl3anc | |
44 | sstr2 | |
|
45 | 43 44 | syl5com | |
46 | topontop | |
|
47 | 46 | ad3antrrr | |
48 | 5 | ad3antrrr | |
49 | 18 48 | eqtrd | |
50 | 16 49 | sseqtrid | |
51 | 8 | clsss3 | |
52 | 47 50 51 | syl2anc | |
53 | 52 49 | sseqtrrd | |
54 | funimass3 | |
|
55 | 36 53 54 | syl2anc | |
56 | 45 55 | sylibd | |
57 | 27 56 | syld | |
58 | 57 | ralrimdva | |
59 | 58 | imdistanda | |
60 | cncls2 | |
|
61 | 59 60 | sylibrd | |
62 | 13 61 | impbid | |