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Mirrors > Home > ILE Home > Th. List > tgcn | Unicode version |
Description: The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
tgcn.1 | TopOn |
tgcn.3 | |
tgcn.4 | TopOn |
Ref | Expression |
---|---|
tgcn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcn.1 | . . 3 TopOn | |
2 | tgcn.4 | . . 3 TopOn | |
3 | iscn 12366 | . . 3 TopOn TopOn | |
4 | 1, 2, 3 | syl2anc 408 | . 2 |
5 | tgcn.3 | . . . . . . . . 9 | |
6 | topontop 12181 | . . . . . . . . . 10 TopOn | |
7 | 2, 6 | syl 14 | . . . . . . . . 9 |
8 | 5, 7 | eqeltrrd 2217 | . . . . . . . 8 |
9 | tgclb 12234 | . . . . . . . 8 | |
10 | 8, 9 | sylibr 133 | . . . . . . 7 |
11 | bastg 12230 | . . . . . . 7 | |
12 | 10, 11 | syl 14 | . . . . . 6 |
13 | 12, 5 | sseqtrrd 3136 | . . . . 5 |
14 | ssralv 3161 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 5 | eleq2d 2209 | . . . . . . . . 9 |
17 | eltg3 12226 | . . . . . . . . . 10 | |
18 | 10, 17 | syl 14 | . . . . . . . . 9 |
19 | 16, 18 | bitrd 187 | . . . . . . . 8 |
20 | ssralv 3161 | . . . . . . . . . . . 12 | |
21 | topontop 12181 | . . . . . . . . . . . . . 14 TopOn | |
22 | 1, 21 | syl 14 | . . . . . . . . . . . . 13 |
23 | iunopn 12169 | . . . . . . . . . . . . . 14 | |
24 | 23 | ex 114 | . . . . . . . . . . . . 13 |
25 | 22, 24 | syl 14 | . . . . . . . . . . . 12 |
26 | 20, 25 | sylan9r 407 | . . . . . . . . . . 11 |
27 | imaeq2 4877 | . . . . . . . . . . . . . 14 | |
28 | imauni 5662 | . . . . . . . . . . . . . 14 | |
29 | 27, 28 | syl6eq 2188 | . . . . . . . . . . . . 13 |
30 | 29 | eleq1d 2208 | . . . . . . . . . . . 12 |
31 | 30 | imbi2d 229 | . . . . . . . . . . 11 |
32 | 26, 31 | syl5ibrcom 156 | . . . . . . . . . 10 |
33 | 32 | expimpd 360 | . . . . . . . . 9 |
34 | 33 | exlimdv 1791 | . . . . . . . 8 |
35 | 19, 34 | sylbid 149 | . . . . . . 7 |
36 | 35 | imp 123 | . . . . . 6 |
37 | 36 | ralrimdva 2512 | . . . . 5 |
38 | imaeq2 4877 | . . . . . . 7 | |
39 | 38 | eleq1d 2208 | . . . . . 6 |
40 | 39 | cbvralv 2654 | . . . . 5 |
41 | 37, 40 | syl6ib 160 | . . . 4 |
42 | 15, 41 | impbid 128 | . . 3 |
43 | 42 | anbi2d 459 | . 2 |
44 | 4, 43 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 wss 3071 cuni 3736 ciun 3813 ccnv 4538 cima 4542 wf 5119 cfv 5123 (class class class)co 5774 ctg 12135 ctop 12164 TopOnctopon 12177 ctb 12209 ccn 12354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 |
This theorem is referenced by: txcnmpt 12442 |
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