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Mirrors > Home > ILE Home > Th. List > ntrval | 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 12641 | . . . 4 |
3 | 2 | fveq1d 5482 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | eqid 2164 | . . 3 | |
6 | pweq 3556 | . . . . 5 | |
7 | 6 | ineq2d 3318 | . . . 4 |
8 | 7 | unieqd 3794 | . . 3 |
9 | 1 | topopn 12547 | . . . . 5 |
10 | elpw2g 4129 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 11 | biimpar 295 | . . 3 |
13 | inex1g 4112 | . . . . 5 | |
14 | 13 | adantr 274 | . . . 4 |
15 | uniexg 4411 | . . . 4 | |
16 | 14, 15 | syl 14 | . . 3 |
17 | 5, 8, 12, 16 | fvmptd3 5573 | . 2 |
18 | 4, 17 | eqtrd 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 cin 3110 wss 3111 cpw 3553 cuni 3783 cmpt 4037 cfv 5182 ctop 12536 cnt 12634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-top 12537 df-ntr 12637 |
This theorem is referenced by: ntropn 12658 ntrss 12660 ntrss2 12662 ssntr 12663 isopn3 12666 ntreq0 12673 |
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