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Mirrors > Home > MPE Home > Th. List > rncmp | Structured version Visualization version Unicode version |
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
rncmp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 464 |
. 2
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2 | eqid 2471 |
. . . . . . 7
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3 | eqid 2471 |
. . . . . . 7
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4 | 2, 3 | cnf 20339 |
. . . . . 6
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5 | 4 | adantl 473 |
. . . . 5
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6 | ffn 5739 |
. . . . 5
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7 | 5, 6 | syl 17 |
. . . 4
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8 | dffn4 5812 |
. . . 4
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9 | 7, 8 | sylib 201 |
. . 3
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10 | cntop2 20334 |
. . . . . 6
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11 | 10 | adantl 473 |
. . . . 5
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12 | frn 5747 |
. . . . . 6
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13 | 5, 12 | syl 17 |
. . . . 5
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14 | 3 | restuni 20255 |
. . . . 5
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15 | 11, 13, 14 | syl2anc 673 |
. . . 4
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16 | foeq3 5804 |
. . . 4
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17 | 15, 16 | syl 17 |
. . 3
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18 | 9, 17 | mpbid 215 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | simpr 468 |
. . 3
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20 | 3 | toptopon 20025 |
. . . . 5
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21 | 11, 20 | sylib 201 |
. . . 4
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22 | ssid 3437 |
. . . . 5
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23 | 22 | a1i 11 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | cnrest2 20379 |
. . . 4
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25 | 21, 23, 13, 24 | syl3anc 1292 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 19, 25 | mpbid 215 |
. 2
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27 | eqid 2471 |
. . 3
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28 | 27 | cncmp 20484 |
. 2
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29 | 1, 18, 26, 28 | syl3anc 1292 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-oadd 7204 df-er 7381 df-map 7492 df-en 7588 df-dom 7589 df-fin 7591 df-fi 7943 df-rest 15399 df-topgen 15420 df-top 19998 df-bases 19999 df-topon 20000 df-cn 20320 df-cmp 20479 |
This theorem is referenced by: imacmp 20489 kgencn2 20649 bndth 22064 |
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