Step | Hyp | Ref
| Expression |
1 | | relcmpcmet.2 |
. 2
       |
2 | | metxmet 21349 |
. . . . . . 7
    
       |
3 | 1, 2 | syl 17 |
. . . . . 6
        |
4 | 3 | adantr 467 |
. . . . 5
 
CauFil          |
5 | | simpr 463 |
. . . . 5
 
CauFil   CauFil    |
6 | | relcmpcmet.3 |
. . . . . 6
   |
7 | 6 | adantr 467 |
. . . . 5
 
CauFil     |
8 | | cfil3i 22239 |
. . . . 5
       CauFil   
          |
9 | 4, 5, 7, 8 | syl3anc 1268 |
. . . 4
 
CauFil              |
10 | 3 | ad2antrr 732 |
. . . . . . . 8
   CauFil  

         
       |
11 | | relcmpcmet.1 |
. . . . . . . . 9
     |
12 | 11 | mopntopon 21454 |
. . . . . . . 8
      TopOn    |
13 | 10, 12 | syl 17 |
. . . . . . 7
   CauFil  

         
TopOn    |
14 | | cfilfil 22237 |
. . . . . . . . 9
       CauFil         |
15 | 3, 14 | sylan 474 |
. . . . . . . 8
 
CauFil         |
16 | 15 | adantr 467 |
. . . . . . 7
   CauFil  

                |
17 | | simprr 766 |
. . . . . . . 8
   CauFil  

                    |
18 | | topontop 19941 |
. . . . . . . . . . 11
 TopOn 
  |
19 | 13, 18 | syl 17 |
. . . . . . . . . 10
   CauFil  

         
  |
20 | | simprl 764 |
. . . . . . . . . . . 12
   CauFil  

         
  |
21 | 6 | rpxrd 11342 |
. . . . . . . . . . . . 13
   |
22 | 21 | ad2antrr 732 |
. . . . . . . . . . . 12
   CauFil  

         
  |
23 | | blssm 21433 |
. . . . . . . . . . . 12
      
           |
24 | 10, 20, 22, 23 | syl3anc 1268 |
. . . . . . . . . . 11
   CauFil  

                    |
25 | | toponuni 19942 |
. . . . . . . . . . . 12
 TopOn 
   |
26 | 13, 25 | syl 17 |
. . . . . . . . . . 11
   CauFil  

         
   |
27 | 24, 26 | sseqtrd 3468 |
. . . . . . . . . 10
   CauFil  

                     |
28 | | eqid 2451 |
. . . . . . . . . . 11
   |
29 | 28 | clsss3 20074 |
. . . . . . . . . 10
                               |
30 | 19, 27, 29 | syl2anc 667 |
. . . . . . . . 9
   CauFil  

                             |
31 | 30, 26 | sseqtr4d 3469 |
. . . . . . . 8
   CauFil  

                            |
32 | 28 | sscls 20071 |
. . . . . . . . 9
                                      |
33 | 19, 27, 32 | syl2anc 667 |
. . . . . . . 8
   CauFil  

                                    |
34 | | filss 20868 |
. . . . . . . 8
                                                        
                  |
35 | 16, 17, 31, 33, 34 | syl13anc 1270 |
. . . . . . 7
   CauFil  

                            |
36 | | fclsrest 21039 |
. . . . . . 7
  TopOn     
                   ↾t                  
↾t                                         |
37 | 13, 16, 35, 36 | syl3anc 1268 |
. . . . . 6
   CauFil  

            ↾t                  
↾t                                         |
38 | | inss1 3652 |
. . . . . . 7
 
                     |
39 | | eqid 2451 |
. . . . . . . . 9
 |
40 | 11, 39 | cfilfcls 22244 |
. . . . . . . 8
 CauFil 
  
   |
41 | 40 | ad2antlr 733 |
. . . . . . 7
   CauFil  

                |
42 | 38, 41 | syl5sseq 3480 |
. . . . . 6
   CauFil  

                                  |
43 | 37, 42 | eqsstrd 3466 |
. . . . 5
   CauFil  

            ↾t                  
↾t                       |
44 | | relcmpcmet.4 |
. . . . . . 7
 
  ↾t                    |
45 | 44 | ad2ant2r 753 |
. . . . . 6
   CauFil  

          
↾t                    |
46 | | filfbas 20863 |
. . . . . . . . . 10
    
      |
47 | 16, 46 | syl 17 |
. . . . . . . . 9
   CauFil  

                |
48 | | fbncp 20854 |
. . . . . . . . 9
                       
                   |
49 | 47, 35, 48 | syl2anc 667 |
. . . . . . . 8
   CauFil  

                              |
50 | | trfil3 20903 |
. . . . . . . . 9
                        
↾t                                                           |
51 | 16, 31, 50 | syl2anc 667 |
. . . . . . . 8
   CauFil  

            ↾t                                                           |
52 | 49, 51 | mpbird 236 |
. . . . . . 7
   CauFil  

          
↾t                                        |
53 | | resttopon 20177 |
. . . . . . . . . 10
  TopOn                    ↾t                  TopOn                    |
54 | 13, 31, 53 | syl2anc 667 |
. . . . . . . . 9
   CauFil  

          
↾t                  TopOn                    |
55 | | toponuni 19942 |
. . . . . . . . 9
  ↾t                  TopOn                                    ↾t                    |
56 | 54, 55 | syl 17 |
. . . . . . . 8
   CauFil  

                            ↾t                    |
57 | 56 | fveq2d 5869 |
. . . . . . 7
   CauFil  

                                  
↾t                     |
58 | 52, 57 | eleqtrd 2531 |
. . . . . 6
   CauFil  

          
↾t                       ↾t                     |
59 | | eqid 2451 |
. . . . . . 7
  ↾t                    ↾t                   |
60 | 59 | fclscmpi 21044 |
. . . . . 6
  
↾t                   ↾t                      
↾t                      ↾t                  
↾t                     |
61 | 45, 58, 60 | syl2anc 667 |
. . . . 5
   CauFil  

            ↾t                  
↾t                     |
62 | | ssn0 3767 |
. . . . 5
    ↾t                  
↾t                       ↾t                  
↾t                        |
63 | 43, 61, 62 | syl2anc 667 |
. . . 4
   CauFil  

              |
64 | 9, 63 | rexlimddv 2883 |
. . 3
 
CauFil       |
65 | 64 | ralrimiva 2802 |
. 2
  CauFil       |
66 | 11 | iscmet 22254 |
. 2
         
 CauFil        |
67 | 1, 65, 66 | sylanbrc 670 |
1
       |