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Mirrors > Home > MPE Home > Th. List > isfcls | Structured version Visualization version Unicode version |
Description: A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclsval.x |
Ref | Expression |
---|---|
isfcls |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 681 | . 2 | |
2 | fvssunirn 6217 | . . . . . . . 8 | |
3 | 2 | sseli 3599 | . . . . . . 7 |
4 | filunibas 21685 | . . . . . . . 8 | |
5 | 4 | eqcomd 2628 | . . . . . . 7 |
6 | 3, 5 | jca 554 | . . . . . 6 |
7 | filunirn 21686 | . . . . . . 7 | |
8 | fveq2 6191 | . . . . . . . . 9 | |
9 | 8 | eleq2d 2687 | . . . . . . . 8 |
10 | 9 | biimparc 504 | . . . . . . 7 |
11 | 7, 10 | sylanb 489 | . . . . . 6 |
12 | 6, 11 | impbii 199 | . . . . 5 |
13 | 12 | anbi2i 730 | . . . 4 |
14 | 13 | anbi1i 731 | . . 3 |
15 | df-3an 1039 | . . 3 | |
16 | anass 681 | . . . 4 | |
17 | 16 | anbi1i 731 | . . 3 |
18 | 14, 15, 17 | 3bitr4i 292 | . 2 |
19 | df-fcls 21745 | . . . 4 | |
20 | 19 | elmpt2cl 6876 | . . 3 |
21 | fclsval.x | . . . . . . 7 | |
22 | 21 | fclsval 21812 | . . . . . 6 |
23 | 7, 22 | sylan2b 492 | . . . . 5 |
24 | 23 | eleq2d 2687 | . . . 4 |
25 | n0i 3920 | . . . . . . 7 | |
26 | iffalse 4095 | . . . . . . 7 | |
27 | 25, 26 | nsyl2 142 | . . . . . 6 |
28 | 27 | a1i 11 | . . . . 5 |
29 | 28 | pm4.71rd 667 | . . . 4 |
30 | iftrue 4092 | . . . . . . . 8 | |
31 | 30 | adantl 482 | . . . . . . 7 |
32 | 31 | eleq2d 2687 | . . . . . 6 |
33 | elex 3212 | . . . . . . . 8 | |
34 | 33 | a1i 11 | . . . . . . 7 |
35 | filn0 21666 | . . . . . . . . . . 11 | |
36 | 7, 35 | sylbi 207 | . . . . . . . . . 10 |
37 | 36 | ad2antlr 763 | . . . . . . . . 9 |
38 | r19.2z 4060 | . . . . . . . . . 10 | |
39 | 38 | ex 450 | . . . . . . . . 9 |
40 | 37, 39 | syl 17 | . . . . . . . 8 |
41 | elex 3212 | . . . . . . . . 9 | |
42 | 41 | rexlimivw 3029 | . . . . . . . 8 |
43 | 40, 42 | syl6 35 | . . . . . . 7 |
44 | eliin 4525 | . . . . . . . 8 | |
45 | 44 | a1i 11 | . . . . . . 7 |
46 | 34, 43, 45 | pm5.21ndd 369 | . . . . . 6 |
47 | 32, 46 | bitrd 268 | . . . . 5 |
48 | 47 | pm5.32da 673 | . . . 4 |
49 | 24, 29, 48 | 3bitrd 294 | . . 3 |
50 | 20, 49 | biadan2 674 | . 2 |
51 | 1, 18, 50 | 3bitr4ri 293 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 wrex 2913 cvv 3200 c0 3915 cif 4086 cuni 4436 ciin 4521 crn 5115 cfv 5888 (class class class)co 6650 ctop 20698 ccl 20822 cfil 21649 cfcls 21740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-fbas 19743 df-fil 21650 df-fcls 21745 |
This theorem is referenced by: fclsfil 21814 fclstop 21815 isfcls2 21817 fclssscls 21822 flimfcls 21830 |
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