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Mirrors > Home > MPE Home > Th. List > restperf | Structured version Visualization version Unicode version |
Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
restcls.1 |
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restcls.2 |
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Ref | Expression |
---|---|
restperf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restcls.2 |
. . . . 5
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2 | restcls.1 |
. . . . . . 7
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3 | 2 | toptopon 19948 |
. . . . . 6
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4 | resttopon 20177 |
. . . . . 6
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5 | 3, 4 | sylanb 475 |
. . . . 5
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6 | 1, 5 | syl5eqel 2533 |
. . . 4
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7 | topontop 19941 |
. . . 4
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8 | 6, 7 | syl 17 |
. . 3
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9 | eqid 2451 |
. . . . 5
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10 | 9 | isperf 20167 |
. . . 4
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11 | 10 | baib 914 |
. . 3
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12 | 8, 11 | syl 17 |
. 2
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13 | dfss1 3637 |
. . 3
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14 | ssid 3451 |
. . . . . 6
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15 | 2, 1 | restlp 20199 |
. . . . . 6
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16 | 14, 15 | mp3an3 1353 |
. . . . 5
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17 | toponuni 19942 |
. . . . . . 7
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18 | 6, 17 | syl 17 |
. . . . . 6
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19 | 18 | fveq2d 5869 |
. . . . 5
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20 | 16, 19 | eqtr3d 2487 |
. . . 4
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21 | 20, 18 | eqeq12d 2466 |
. . 3
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22 | 13, 21 | syl5bb 261 |
. 2
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23 | 12, 22 | bitr4d 260 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-oadd 7186 df-er 7363 df-en 7570 df-fin 7573 df-fi 7925 df-rest 15321 df-topgen 15342 df-top 19921 df-bases 19922 df-topon 19923 df-cld 20034 df-cls 20036 df-lp 20152 df-perf 20153 |
This theorem is referenced by: perfcls 20381 reperflem 21836 perfdvf 22858 |
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