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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 12360 | . . . . . . . 8 ↾t ↾t | |
3 | 2 | adantl 275 | . . . . . . 7 ↾t ↾t |
4 | restrcl 12325 | . . . . . . 7 ↾t | |
5 | eqid 2137 | . . . . . . . 8 | |
6 | 5 | restin 12334 | . . . . . . 7 ↾t ↾t |
7 | 3, 4, 6 | 3syl 17 | . . . . . 6 ↾t ↾t ↾t |
8 | 7 | oveq2d 5783 | . . . . 5 ↾t ↾t ↾t |
9 | 1, 8 | eleqtrd 2216 | . . . 4 ↾t ↾t |
10 | simpl 108 | . . . . . 6 ↾t | |
11 | 5 | toptopon 12174 | . . . . . 6 TopOn |
12 | 10, 11 | sylib 121 | . . . . 5 ↾t TopOn |
13 | cntop1 12359 | . . . . . . . . 9 ↾t | |
14 | 13 | adantl 275 | . . . . . . . 8 ↾t |
15 | eqid 2137 | . . . . . . . . 9 | |
16 | 15 | toptopon 12174 | . . . . . . . 8 TopOn |
17 | 14, 16 | sylib 121 | . . . . . . 7 ↾t TopOn |
18 | inss2 3292 | . . . . . . . 8 | |
19 | resttopon 12329 | . . . . . . . 8 TopOn ↾t TopOn | |
20 | 12, 18, 19 | sylancl 409 | . . . . . . 7 ↾t ↾t TopOn |
21 | cnf2 12363 | . . . . . . 7 TopOn ↾t TopOn ↾t | |
22 | 17, 20, 9, 21 | syl3anc 1216 | . . . . . 6 ↾t |
23 | 22 | frnd 5277 | . . . . 5 ↾t |
24 | 18 | a1i 9 | . . . . 5 ↾t |
25 | cnrest2 12394 | . . . . 5 TopOn ↾t | |
26 | 12, 23, 24, 25 | syl3anc 1216 | . . . 4 ↾t ↾t |
27 | 9, 26 | mpbird 166 | . . 3 ↾t |
28 | 27 | ex 114 | . 2 ↾t |
29 | 28 | ssrdv 3098 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2681 cin 3065 wss 3066 cuni 3731 crn 4535 wf 5114 cfv 5118 (class class class)co 5767 ↾t crest 12109 ctop 12153 TopOnctopon 12166 ccn 12343 |
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 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-rest 12111 df-topgen 12130 df-top 12154 df-topon 12167 df-bases 12199 df-cn 12346 |
This theorem is referenced by: cnrehmeocntop 12751 |
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