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Mirrors > Home > MPE Home > Th. List > clsval | Structured version Visualization version 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 |
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Ref | Expression |
---|---|
clsval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 |
. . . . 5
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2 | 1 | clsfval 20089 |
. . . 4
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3 | 2 | fveq1d 5890 |
. . 3
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4 | 3 | adantr 471 |
. 2
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5 | 1 | topopn 19985 |
. . . . 5
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6 | elpw2g 4580 |
. . . . 5
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7 | 5, 6 | syl 17 |
. . . 4
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8 | 7 | biimpar 492 |
. . 3
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9 | 1 | topcld 20099 |
. . . . 5
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10 | sseq2 3466 |
. . . . . 6
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11 | 10 | rspcev 3162 |
. . . . 5
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12 | 9, 11 | sylan 478 |
. . . 4
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13 | intexrab 4576 |
. . . 4
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14 | 12, 13 | sylib 201 |
. . 3
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15 | sseq1 3465 |
. . . . . 6
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16 | 15 | rabbidv 3048 |
. . . . 5
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17 | 16 | inteqd 4253 |
. . . 4
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18 | eqid 2462 |
. . . 4
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19 | 17, 18 | fvmptg 5969 |
. . 3
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20 | 8, 14, 19 | syl2anc 671 |
. 2
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21 | 4, 20 | eqtrd 2496 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-top 19970 df-cld 20083 df-cls 20085 |
This theorem is referenced by: cldcls 20106 clscld 20111 clsf 20112 clsval2 20114 clsss 20118 sscls 20120 |
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