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Mirrors > Home > ILE Home > Th. List > cnclima | Unicode version |
Description: A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnclima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2100 |
. . . . . 6
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2 | eqid 2100 |
. . . . . 6
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3 | 1, 2 | cnf 12154 |
. . . . 5
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4 | 3 | adantr 272 |
. . . 4
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5 | ffun 5211 |
. . . . . 6
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6 | funcnvcnv 5118 |
. . . . . 6
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7 | imadif 5139 |
. . . . . 6
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8 | 5, 6, 7 | 3syl 17 |
. . . . 5
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9 | fimacnv 5481 |
. . . . . 6
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10 | 9 | difeq1d 3140 |
. . . . 5
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11 | 8, 10 | eqtr2d 2133 |
. . . 4
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12 | 4, 11 | syl 14 |
. . 3
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13 | 2 | cldopn 12058 |
. . . 4
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14 | cnima 12170 |
. . . 4
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15 | 13, 14 | sylan2 282 |
. . 3
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16 | 12, 15 | eqeltrd 2176 |
. 2
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17 | cntop1 12151 |
. . . 4
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18 | 17 | adantr 272 |
. . 3
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19 | cnvimass 4838 |
. . . 4
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20 | 19, 4 | fssdm 5223 |
. . 3
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21 | 1 | iscld2 12055 |
. . 3
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22 | 18, 20, 21 | syl2anc 406 |
. 2
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23 | 16, 22 | mpbird 166 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-map 6474 df-top 11947 df-topon 11960 df-cld 12046 df-cn 12139 |
This theorem is referenced by: (None) |
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