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Mirrors > Home > MPE Home > Th. List > cndis | Structured version Visualization version Unicode version |
Description: Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cndis |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 5206 |
. . . . . . . 8
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2 | fdm 5755 |
. . . . . . . . 9
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3 | 2 | adantl 472 |
. . . . . . . 8
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4 | 1, 3 | syl5sseq 3491 |
. . . . . . 7
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5 | elpw2g 4579 |
. . . . . . . 8
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6 | 5 | ad2antrr 737 |
. . . . . . 7
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7 | 4, 6 | mpbird 240 |
. . . . . 6
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8 | 7 | ralrimivw 2814 |
. . . . 5
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9 | 8 | ex 440 |
. . . 4
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10 | 9 | pm4.71d 644 |
. . 3
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11 | toponmax 19991 |
. . . 4
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12 | id 22 |
. . . 4
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13 | elmapg 7510 |
. . . 4
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14 | 11, 12, 13 | syl2anr 485 |
. . 3
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15 | distopon 20060 |
. . . 4
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16 | iscn 20299 |
. . . 4
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17 | 15, 16 | sylan 478 |
. . 3
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18 | 10, 14, 17 | 3bitr4rd 294 |
. 2
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19 | 18 | eqrdv 2459 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-mpt 4476 df-id 4767 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-fv 5608 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-map 7499 df-top 19969 df-topon 19971 df-cn 20291 |
This theorem is referenced by: xkopt 20718 distgp 21162 symgtgp 21164 |
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