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Mirrors > Home > ILE Home > Th. List > clsval | Unicode version |
Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
clsval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 | |
2 | 1 | clsfval 12270 | . . . 4 |
3 | 2 | fveq1d 5423 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | eqid 2139 | . . 3 | |
6 | sseq1 3120 | . . . . 5 | |
7 | 6 | rabbidv 2675 | . . . 4 |
8 | 7 | inteqd 3776 | . . 3 |
9 | 1 | topopn 12175 | . . . . 5 |
10 | elpw2g 4081 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 11 | biimpar 295 | . . 3 |
13 | 1 | topcld 12278 | . . . . 5 |
14 | sseq2 3121 | . . . . . 6 | |
15 | 14 | rspcev 2789 | . . . . 5 |
16 | 13, 15 | sylan 281 | . . . 4 |
17 | intexrabim 4078 | . . . 4 | |
18 | 16, 17 | syl 14 | . . 3 |
19 | 5, 8, 12, 18 | fvmptd3 5514 | . 2 |
20 | 4, 19 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 crab 2420 cvv 2686 wss 3071 cpw 3510 cuni 3736 cint 3771 cmpt 3989 cfv 5123 ctop 12164 ccld 12261 ccl 12263 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-cld 12264 df-cls 12266 |
This theorem is referenced by: cldcls 12283 clsss 12287 sscls 12289 |
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