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Mirrors > Home > MPE Home > Th. List > flfval | Structured version Unicode version |
Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
flfval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponmax 18675 |
. . . . 5
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2 | filtop 19570 |
. . . . 5
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3 | elmapg 7340 |
. . . . 5
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4 | 1, 2, 3 | syl2an 477 |
. . . 4
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5 | 4 | biimpar 485 |
. . 3
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6 | flffval 19704 |
. . . . 5
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7 | 6 | fveq1d 5804 |
. . . 4
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8 | oveq2 6211 |
. . . . . . 7
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9 | 8 | fveq1d 5804 |
. . . . . 6
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10 | 9 | oveq2d 6219 |
. . . . 5
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11 | eqid 2454 |
. . . . 5
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12 | ovex 6228 |
. . . . 5
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13 | 10, 11, 12 | fvmpt 5886 |
. . . 4
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14 | 7, 13 | sylan9eq 2515 |
. . 3
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15 | 5, 14 | syldan 470 |
. 2
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16 | 15 | 3impa 1183 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-map 7329 df-fbas 17949 df-top 18645 df-topon 18648 df-fil 19561 df-flf 19655 |
This theorem is referenced by: flfnei 19706 isflf 19708 hausflf 19712 flfcnp 19719 flfssfcf 19753 uffcfflf 19754 cnpfcf 19756 cnextcn 19781 tsmscls 19850 cnextucn 20020 cmetcaulem 20941 fmcncfil 26529 |
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