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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 2697 | . . 3 | |
2 | elrest 12130 | . . . . . . 7 ↾t | |
3 | elrest 12130 | . . . . . . 7 ↾t | |
4 | 2, 3 | anbi12d 464 | . . . . . 6 ↾t ↾t |
5 | reeanv 2600 | . . . . . 6 | |
6 | 4, 5 | syl6bbr 197 | . . . . 5 ↾t ↾t |
7 | simplll 522 | . . . . . . . . . 10 | |
8 | simplrl 524 | . . . . . . . . . 10 | |
9 | simplrr 525 | . . . . . . . . . 10 | |
10 | simpr 109 | . . . . . . . . . . 11 | |
11 | 10 | elin1d 3265 | . . . . . . . . . 10 |
12 | basis2 12218 | . . . . . . . . . 10 | |
13 | 7, 8, 9, 11, 12 | syl22anc 1217 | . . . . . . . . 9 |
14 | simplll 522 | . . . . . . . . . . . 12 | |
15 | 14 | simpld 111 | . . . . . . . . . . 11 |
16 | 14 | simprd 113 | . . . . . . . . . . 11 |
17 | simprl 520 | . . . . . . . . . . 11 | |
18 | elrestr 12131 | . . . . . . . . . . 11 ↾t | |
19 | 15, 16, 17, 18 | syl3anc 1216 | . . . . . . . . . 10 ↾t |
20 | simprrl 528 | . . . . . . . . . . 11 | |
21 | simplr 519 | . . . . . . . . . . . 12 | |
22 | 21 | elin2d 3266 | . . . . . . . . . . 11 |
23 | 20, 22 | elind 3261 | . . . . . . . . . 10 |
24 | simprrr 529 | . . . . . . . . . . 11 | |
25 | 24 | ssrind 3303 | . . . . . . . . . 10 |
26 | eleq2 2203 | . . . . . . . . . . . 12 | |
27 | sseq1 3120 | . . . . . . . . . . . 12 | |
28 | 26, 27 | anbi12d 464 | . . . . . . . . . . 11 |
29 | 28 | rspcev 2789 | . . . . . . . . . 10 ↾t ↾t |
30 | 19, 23, 25, 29 | syl12anc 1214 | . . . . . . . . 9 ↾t |
31 | 13, 30 | rexlimddv 2554 | . . . . . . . 8 ↾t |
32 | 31 | ralrimiva 2505 | . . . . . . 7 ↾t |
33 | ineq12 3272 | . . . . . . . . 9 | |
34 | inindir 3294 | . . . . . . . . 9 | |
35 | 33, 34 | syl6eqr 2190 | . . . . . . . 8 |
36 | 35 | sseq2d 3127 | . . . . . . . . . 10 |
37 | 36 | anbi2d 459 | . . . . . . . . 9 |
38 | 37 | rexbidv 2438 | . . . . . . . 8 ↾t ↾t |
39 | 35, 38 | raleqbidv 2638 | . . . . . . 7 ↾t ↾t |
40 | 32, 39 | syl5ibrcom 156 | . . . . . 6 ↾t |
41 | 40 | rexlimdvva 2557 | . . . . 5 ↾t |
42 | 6, 41 | sylbid 149 | . . . 4 ↾t ↾t ↾t |
43 | 42 | ralrimivv 2513 | . . 3 ↾t ↾t ↾t |
44 | 1, 43 | sylan2 284 | . 2 ↾t ↾t ↾t |
45 | restfn 12127 | . . . 4 ↾t | |
46 | simpl 108 | . . . . 5 | |
47 | 46 | elexd 2699 | . . . 4 |
48 | 1 | adantl 275 | . . . 4 |
49 | fnovex 5804 | . . . 4 ↾t ↾t | |
50 | 45, 47, 48, 49 | mp3an2i 1320 | . . 3 ↾t |
51 | isbasis2g 12215 | . . 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 1331 wcel 1480 wral 2416 wrex 2417 cvv 2686 cin 3070 wss 3071 cxp 4537 wfn 5118 (class class class)co 5774 ↾t crest 12123 ctb 12212 |
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-coll 4043 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-reu 2423 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-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-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-rest 12125 df-bases 12213 |
This theorem is referenced by: resttop 12342 |
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