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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 12744 | . . 3 TopOn TopOn | |
4 | 1, 2, 3 | syl2anc 409 | . 2 |
5 | tgcn.3 | . . . . . . . . 9 | |
6 | topontop 12559 | . . . . . . . . . 10 TopOn | |
7 | 2, 6 | syl 14 | . . . . . . . . 9 |
8 | 5, 7 | eqeltrrd 2242 | . . . . . . . 8 |
9 | tgclb 12612 | . . . . . . . 8 | |
10 | 8, 9 | sylibr 133 | . . . . . . 7 |
11 | bastg 12608 | . . . . . . 7 | |
12 | 10, 11 | syl 14 | . . . . . 6 |
13 | 12, 5 | sseqtrrd 3176 | . . . . 5 |
14 | ssralv 3201 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 5 | eleq2d 2234 | . . . . . . . . 9 |
17 | eltg3 12604 | . . . . . . . . . 10 | |
18 | 10, 17 | syl 14 | . . . . . . . . 9 |
19 | 16, 18 | bitrd 187 | . . . . . . . 8 |
20 | ssralv 3201 | . . . . . . . . . . . 12 | |
21 | topontop 12559 | . . . . . . . . . . . . . 14 TopOn | |
22 | 1, 21 | syl 14 | . . . . . . . . . . . . 13 |
23 | iunopn 12547 | . . . . . . . . . . . . . 14 | |
24 | 23 | ex 114 | . . . . . . . . . . . . 13 |
25 | 22, 24 | syl 14 | . . . . . . . . . . . 12 |
26 | 20, 25 | sylan9r 408 | . . . . . . . . . . 11 |
27 | imaeq2 4936 | . . . . . . . . . . . . . 14 | |
28 | imauni 5723 | . . . . . . . . . . . . . 14 | |
29 | 27, 28 | eqtrdi 2213 | . . . . . . . . . . . . 13 |
30 | 29 | eleq1d 2233 | . . . . . . . . . . . 12 |
31 | 30 | imbi2d 229 | . . . . . . . . . . 11 |
32 | 26, 31 | syl5ibrcom 156 | . . . . . . . . . 10 |
33 | 32 | expimpd 361 | . . . . . . . . 9 |
34 | 33 | exlimdv 1806 | . . . . . . . 8 |
35 | 19, 34 | sylbid 149 | . . . . . . 7 |
36 | 35 | imp 123 | . . . . . 6 |
37 | 36 | ralrimdva 2544 | . . . . 5 |
38 | imaeq2 4936 | . . . . . . 7 | |
39 | 38 | eleq1d 2233 | . . . . . 6 |
40 | 39 | cbvralv 2689 | . . . . 5 |
41 | 37, 40 | syl6ib 160 | . . . 4 |
42 | 15, 41 | impbid 128 | . . 3 |
43 | 42 | anbi2d 460 | . 2 |
44 | 4, 43 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 wral 2442 wss 3111 cuni 3783 ciun 3860 ccnv 4597 cima 4601 wf 5178 cfv 5182 (class class class)co 5836 ctg 12513 ctop 12542 TopOnctopon 12555 ctb 12587 ccn 12732 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cn 12735 |
This theorem is referenced by: txcnmpt 12820 |
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