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Mirrors > Home > MPE Home > Th. List > cnflf | Structured version Visualization version Unicode version |
Description: A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
Ref | Expression |
---|---|
cnflf | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncnp 21084 | . 2 TopOn TopOn | |
2 | cnpflf 21805 | . . . . . . . 8 TopOn TopOn | |
3 | 2 | 3expa 1265 | . . . . . . 7 TopOn TopOn |
4 | 3 | adantlr 751 | . . . . . 6 TopOn TopOn |
5 | simplr 792 | . . . . . . 7 TopOn TopOn | |
6 | 5 | biantrurd 529 | . . . . . 6 TopOn TopOn |
7 | 4, 6 | bitr4d 271 | . . . . 5 TopOn TopOn |
8 | 7 | ralbidva 2985 | . . . 4 TopOn TopOn |
9 | eqid 2622 | . . . . . . . . . . . 12 | |
10 | 9 | flimelbas 21772 | . . . . . . . . . . 11 |
11 | toponuni 20719 | . . . . . . . . . . . . 13 TopOn | |
12 | 11 | ad2antrr 762 | . . . . . . . . . . . 12 TopOn TopOn |
13 | 12 | eleq2d 2687 | . . . . . . . . . . 11 TopOn TopOn |
14 | 10, 13 | syl5ibr 236 | . . . . . . . . . 10 TopOn TopOn |
15 | 14 | pm4.71rd 667 | . . . . . . . . 9 TopOn TopOn |
16 | 15 | imbi1d 331 | . . . . . . . 8 TopOn TopOn |
17 | impexp 462 | . . . . . . . 8 | |
18 | 16, 17 | syl6bb 276 | . . . . . . 7 TopOn TopOn |
19 | 18 | ralbidv2 2984 | . . . . . 6 TopOn TopOn |
20 | 19 | ralbidv 2986 | . . . . 5 TopOn TopOn |
21 | ralcom 3098 | . . . . 5 | |
22 | 20, 21 | syl6bb 276 | . . . 4 TopOn TopOn |
23 | 8, 22 | bitr4d 271 | . . 3 TopOn TopOn |
24 | 23 | pm5.32da 673 | . 2 TopOn TopOn |
25 | 1, 24 | bitrd 268 | 1 TopOn TopOn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cuni 4436 wf 5884 cfv 5888 (class class class)co 6650 TopOnctopon 20715 ccn 21028 ccnp 21029 cfil 21649 cflim 21738 cflf 21739 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-reu 2919 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-iun 4522 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-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-ntr 20824 df-nei 20902 df-cn 21031 df-cnp 21032 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 |
This theorem is referenced by: cnflf2 21807 flfcntr 21847 fmcncfil 29977 |
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