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Mirrors > Home > ILE Home > Th. List > hmeocld | Unicode version |
Description: Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoopn.1 |
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Ref | Expression |
---|---|
hmeocld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnvcn 13668 |
. . . 4
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2 | 1 | adantr 276 |
. . 3
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3 | imacnvcnv 5091 |
. . . . 5
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4 | cnclima 13585 |
. . . . 5
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5 | 3, 4 | eqeltrrid 2265 |
. . . 4
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6 | 5 | ex 115 |
. . 3
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7 | 2, 6 | syl 14 |
. 2
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8 | hmeocn 13667 |
. . . . 5
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9 | 8 | adantr 276 |
. . . 4
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10 | cnclima 13585 |
. . . . 5
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11 | 10 | ex 115 |
. . . 4
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12 | 9, 11 | syl 14 |
. . 3
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13 | hmeoopn.1 |
. . . . . . 7
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14 | eqid 2177 |
. . . . . . 7
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15 | 13, 14 | hmeof1o 13671 |
. . . . . 6
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16 | f1of1 5458 |
. . . . . 6
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17 | 15, 16 | syl 14 |
. . . . 5
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18 | f1imacnv 5476 |
. . . . 5
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19 | 17, 18 | sylan 283 |
. . . 4
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20 | 19 | eleq1d 2246 |
. . 3
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21 | 12, 20 | sylibd 149 |
. 2
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22 | 7, 21 | impbid 129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-map 6646 df-top 13358 df-topon 13371 df-cld 13457 df-cn 13550 df-hmeo 13663 |
This theorem is referenced by: (None) |
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