Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnrest2r | Unicode version |
Description: Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.) |
Ref | Expression |
---|---|
cnrest2r | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 ↾t ↾t | |
2 | cntop2 12743 | . . . . . . . 8 ↾t ↾t | |
3 | 2 | adantl 275 | . . . . . . 7 ↾t ↾t |
4 | restrcl 12708 | . . . . . . 7 ↾t | |
5 | eqid 2164 | . . . . . . . 8 | |
6 | 5 | restin 12717 | . . . . . . 7 ↾t ↾t |
7 | 3, 4, 6 | 3syl 17 | . . . . . 6 ↾t ↾t ↾t |
8 | 7 | oveq2d 5852 | . . . . 5 ↾t ↾t ↾t |
9 | 1, 8 | eleqtrd 2243 | . . . 4 ↾t ↾t |
10 | simpl 108 | . . . . . 6 ↾t | |
11 | 5 | toptopon 12557 | . . . . . 6 TopOn |
12 | 10, 11 | sylib 121 | . . . . 5 ↾t TopOn |
13 | cntop1 12742 | . . . . . . . . 9 ↾t | |
14 | 13 | adantl 275 | . . . . . . . 8 ↾t |
15 | eqid 2164 | . . . . . . . . 9 | |
16 | 15 | toptopon 12557 | . . . . . . . 8 TopOn |
17 | 14, 16 | sylib 121 | . . . . . . 7 ↾t TopOn |
18 | inss2 3338 | . . . . . . . 8 | |
19 | resttopon 12712 | . . . . . . . 8 TopOn ↾t TopOn | |
20 | 12, 18, 19 | sylancl 410 | . . . . . . 7 ↾t ↾t TopOn |
21 | cnf2 12746 | . . . . . . 7 TopOn ↾t TopOn ↾t | |
22 | 17, 20, 9, 21 | syl3anc 1227 | . . . . . 6 ↾t |
23 | 22 | frnd 5341 | . . . . 5 ↾t |
24 | 18 | a1i 9 | . . . . 5 ↾t |
25 | cnrest2 12777 | . . . . 5 TopOn ↾t | |
26 | 12, 23, 24, 25 | syl3anc 1227 | . . . 4 ↾t ↾t |
27 | 9, 26 | mpbird 166 | . . 3 ↾t |
28 | 27 | ex 114 | . 2 ↾t |
29 | 28 | ssrdv 3143 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 cin 3110 wss 3111 cuni 3783 crn 4599 wf 5178 cfv 5182 (class class class)co 5836 ↾t crest 12492 ctop 12536 TopOnctopon 12549 ccn 12726 |
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-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-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-map 6607 df-rest 12494 df-topgen 12513 df-top 12537 df-topon 12550 df-bases 12582 df-cn 12729 |
This theorem is referenced by: cnrehmeocntop 13134 |
Copyright terms: Public domain | W3C validator |