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Mirrors > Home > MPE Home > Th. List > ntrval | Structured version Visualization version Unicode version |
Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
ntrval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 | |
2 | 1 | ntrfval 20828 | . . . 4 |
3 | 2 | fveq1d 6193 | . . 3 |
4 | 3 | adantr 481 | . 2 |
5 | 1 | topopn 20711 | . . . . 5 |
6 | elpw2g 4827 | . . . . 5 | |
7 | 5, 6 | syl 17 | . . . 4 |
8 | 7 | biimpar 502 | . . 3 |
9 | inex1g 4801 | . . . . 5 | |
10 | 9 | adantr 481 | . . . 4 |
11 | uniexg 6955 | . . . 4 | |
12 | 10, 11 | syl 17 | . . 3 |
13 | pweq 4161 | . . . . . 6 | |
14 | 13 | ineq2d 3814 | . . . . 5 |
15 | 14 | unieqd 4446 | . . . 4 |
16 | eqid 2622 | . . . 4 | |
17 | 15, 16 | fvmptg 6280 | . . 3 |
18 | 8, 12, 17 | syl2anc 693 | . 2 |
19 | 4, 18 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cvv 3200 cin 3573 wss 3574 cpw 4158 cuni 4436 cmpt 4729 cfv 5888 ctop 20698 cnt 20821 |
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-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-top 20699 df-ntr 20824 |
This theorem is referenced by: ntropn 20853 clsval2 20854 ntrss2 20861 ssntr 20862 isopn3 20870 ntreq0 20881 |
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