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Mirrors > Home > ILE Home > Th. List > istps | Unicode version |
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istps.a |
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istps.j |
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Ref | Expression |
---|---|
istps |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topsp 12035 |
. . 3
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2 | 1 | eleq2i 2179 |
. 2
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3 | topontop 12018 |
. . . 4
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4 | topnfn 11962 |
. . . . . . 7
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5 | fnrel 5177 |
. . . . . . 7
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6 | 4, 5 | ax-mp 7 |
. . . . . 6
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7 | 0opn 12010 |
. . . . . . 7
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8 | istps.j |
. . . . . . 7
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9 | 7, 8 | syl6eleq 2205 |
. . . . . 6
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10 | relelfvdm 5405 |
. . . . . 6
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11 | 6, 9, 10 | sylancr 408 |
. . . . 5
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12 | 11 | elexd 2668 |
. . . 4
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13 | 3, 12 | syl 14 |
. . 3
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14 | fveq2 5373 |
. . . . 5
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15 | 14, 8 | syl6eqr 2163 |
. . . 4
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16 | fveq2 5373 |
. . . . . 6
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17 | istps.a |
. . . . . 6
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18 | 16, 17 | syl6eqr 2163 |
. . . . 5
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19 | 18 | fveq2d 5377 |
. . . 4
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20 | 15, 19 | eleq12d 2183 |
. . 3
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21 | 13, 20 | elab3 2803 |
. 2
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22 | 2, 21 | bitri 183 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-5 8686 df-6 8687 df-7 8688 df-8 8689 df-9 8690 df-ndx 11799 df-slot 11800 df-base 11802 df-tset 11877 df-rest 11959 df-topn 11960 df-top 12002 df-topon 12015 df-topsp 12035 |
This theorem is referenced by: istps2 12037 tpspropd 12040 tsettps 12042 isxms2 12435 |
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