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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 13692 |
. . 3
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2 | 1 | eleq2i 2244 |
. 2
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3 | topontop 13675 |
. . . 4
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4 | topnfn 12699 |
. . . . . . 7
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5 | fnrel 5316 |
. . . . . . 7
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6 | 4, 5 | ax-mp 5 |
. . . . . 6
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7 | 0opn 13667 |
. . . . . . 7
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8 | istps.j |
. . . . . . 7
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9 | 7, 8 | eleqtrdi 2270 |
. . . . . 6
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10 | relelfvdm 5549 |
. . . . . 6
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11 | 6, 9, 10 | sylancr 414 |
. . . . 5
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12 | 11 | elexd 2752 |
. . . 4
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13 | 3, 12 | syl 14 |
. . 3
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14 | fveq2 5517 |
. . . . 5
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15 | 14, 8 | eqtr4di 2228 |
. . . 4
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16 | fveq2 5517 |
. . . . . 6
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17 | istps.a |
. . . . . 6
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18 | 16, 17 | eqtr4di 2228 |
. . . . 5
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19 | 18 | fveq2d 5521 |
. . . 4
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20 | 15, 19 | eleq12d 2248 |
. . 3
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21 | 13, 20 | elab3 2891 |
. 2
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22 | 2, 21 | bitri 184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-7 8986 df-8 8987 df-9 8988 df-ndx 12468 df-slot 12469 df-base 12471 df-tset 12558 df-rest 12696 df-topn 12697 df-top 13659 df-topon 13672 df-topsp 13692 |
This theorem is referenced by: istps2 13694 tpspropd 13697 tsettps 13699 isxms2 14113 |
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