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Mirrors > Home > ILE Home > Th. List > restbasg | Unicode version |
Description: A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
restbasg | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2732 | . . 3 | |
2 | elrest 12505 | . . . . . . 7 ↾t | |
3 | elrest 12505 | . . . . . . 7 ↾t | |
4 | 2, 3 | anbi12d 465 | . . . . . 6 ↾t ↾t |
5 | reeanv 2633 | . . . . . 6 | |
6 | 4, 5 | bitr4di 197 | . . . . 5 ↾t ↾t |
7 | simplll 523 | . . . . . . . . . 10 | |
8 | simplrl 525 | . . . . . . . . . 10 | |
9 | simplrr 526 | . . . . . . . . . 10 | |
10 | simpr 109 | . . . . . . . . . . 11 | |
11 | 10 | elin1d 3306 | . . . . . . . . . 10 |
12 | basis2 12593 | . . . . . . . . . 10 | |
13 | 7, 8, 9, 11, 12 | syl22anc 1228 | . . . . . . . . 9 |
14 | simplll 523 | . . . . . . . . . . . 12 | |
15 | 14 | simpld 111 | . . . . . . . . . . 11 |
16 | 14 | simprd 113 | . . . . . . . . . . 11 |
17 | simprl 521 | . . . . . . . . . . 11 | |
18 | elrestr 12506 | . . . . . . . . . . 11 ↾t | |
19 | 15, 16, 17, 18 | syl3anc 1227 | . . . . . . . . . 10 ↾t |
20 | simprrl 529 | . . . . . . . . . . 11 | |
21 | simplr 520 | . . . . . . . . . . . 12 | |
22 | 21 | elin2d 3307 | . . . . . . . . . . 11 |
23 | 20, 22 | elind 3302 | . . . . . . . . . 10 |
24 | simprrr 530 | . . . . . . . . . . 11 | |
25 | 24 | ssrind 3344 | . . . . . . . . . 10 |
26 | eleq2 2228 | . . . . . . . . . . . 12 | |
27 | sseq1 3160 | . . . . . . . . . . . 12 | |
28 | 26, 27 | anbi12d 465 | . . . . . . . . . . 11 |
29 | 28 | rspcev 2825 | . . . . . . . . . 10 ↾t ↾t |
30 | 19, 23, 25, 29 | syl12anc 1225 | . . . . . . . . 9 ↾t |
31 | 13, 30 | rexlimddv 2586 | . . . . . . . 8 ↾t |
32 | 31 | ralrimiva 2537 | . . . . . . 7 ↾t |
33 | ineq12 3313 | . . . . . . . . 9 | |
34 | inindir 3335 | . . . . . . . . 9 | |
35 | 33, 34 | eqtr4di 2215 | . . . . . . . 8 |
36 | 35 | sseq2d 3167 | . . . . . . . . . 10 |
37 | 36 | anbi2d 460 | . . . . . . . . 9 |
38 | 37 | rexbidv 2465 | . . . . . . . 8 ↾t ↾t |
39 | 35, 38 | raleqbidv 2671 | . . . . . . 7 ↾t ↾t |
40 | 32, 39 | syl5ibrcom 156 | . . . . . 6 ↾t |
41 | 40 | rexlimdvva 2589 | . . . . 5 ↾t |
42 | 6, 41 | sylbid 149 | . . . 4 ↾t ↾t ↾t |
43 | 42 | ralrimivv 2545 | . . 3 ↾t ↾t ↾t |
44 | 1, 43 | sylan2 284 | . 2 ↾t ↾t ↾t |
45 | restfn 12502 | . . . 4 ↾t | |
46 | simpl 108 | . . . . 5 | |
47 | 46 | elexd 2734 | . . . 4 |
48 | 1 | adantl 275 | . . . 4 |
49 | fnovex 5866 | . . . 4 ↾t ↾t | |
50 | 45, 47, 48, 49 | mp3an2i 1331 | . . 3 ↾t |
51 | isbasis2g 12590 | . . 3 ↾t ↾t ↾t ↾t ↾t | |
52 | 50, 51 | syl 14 | . 2 ↾t ↾t ↾t ↾t |
53 | 44, 52 | mpbird 166 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 cin 3110 wss 3111 cxp 4596 wfn 5177 (class class class)co 5836 ↾t crest 12498 ctb 12587 |
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-coll 4091 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-reu 2449 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-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-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-rest 12500 df-bases 12588 |
This theorem is referenced by: resttop 12717 |
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