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Mirrors > Home > MPE Home > Th. List > kgencmp | Unicode version |
Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgencmp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgenftop 17533 |
. . . 4
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2 | 1 | adantr 452 |
. . 3
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3 | kgenss 17536 |
. . . 4
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4 | 3 | adantr 452 |
. . 3
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5 | ssrest 17202 |
. . 3
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6 | 2, 4, 5 | syl2anc 643 |
. 2
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7 | cmptop 17420 |
. . . . . 6
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8 | 7 | adantl 453 |
. . . . 5
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9 | restrcl 17183 |
. . . . . 6
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10 | 9 | simprd 450 |
. . . . 5
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11 | 8, 10 | syl 16 |
. . . 4
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12 | restval 13617 |
. . . 4
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13 | 2, 11, 12 | syl2anc 643 |
. . 3
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14 | simpr 448 |
. . . . . 6
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15 | simplr 732 |
. . . . . 6
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16 | kgeni 17530 |
. . . . . 6
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17 | 14, 15, 16 | syl2anc 643 |
. . . . 5
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18 | eqid 2412 |
. . . . 5
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19 | 17, 18 | fmptd 5860 |
. . . 4
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20 | frn 5564 |
. . . 4
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21 | 19, 20 | syl 16 |
. . 3
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22 | 13, 21 | eqsstrd 3350 |
. 2
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23 | 6, 22 | eqssd 3333 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: kgencmp2 17539 kgenidm 17540 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-rep 4288 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-ral 2679 df-rex 2680 df-reu 2681 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-pss 3304 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-tp 3790 df-op 3791 df-uni 3984 df-int 4019 df-iun 4063 df-br 4181 df-opab 4235 df-mpt 4236 df-tr 4271 df-eprel 4462 df-id 4466 df-po 4471 df-so 4472 df-fr 4509 df-we 4511 df-ord 4552 df-on 4553 df-lim 4554 df-suc 4555 df-om 4813 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-1st 6316 df-2nd 6317 df-recs 6600 df-rdg 6635 df-oadd 6695 df-er 6872 df-en 7077 df-fin 7080 df-fi 7382 df-rest 13613 df-topgen 13630 df-top 16926 df-bases 16928 df-topon 16929 df-cmp 17412 df-kgen 17527 |
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