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Mirrors > Home > ILE Home > Th. List > cnptoprest2 | Unicode version |
Description: Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.) |
Ref | Expression |
---|---|
cnprest.1 | |
cnprest.2 |
Ref | Expression |
---|---|
cnptoprest2 | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnprest.1 | . . . . . . . 8 | |
2 | 1 | toptopon 12185 | . . . . . . 7 TopOn |
3 | 2 | biimpi 119 | . . . . . 6 TopOn |
4 | 3 | ad2antrr 479 | . . . . 5 TopOn |
5 | 4 | adantr 274 | . . . 4 TopOn |
6 | simplr 519 | . . . . 5 | |
7 | 6 | adantr 274 | . . . 4 |
8 | simpr 109 | . . . 4 | |
9 | cnprcl2k 12375 | . . . 4 TopOn | |
10 | 5, 7, 8, 9 | syl3anc 1216 | . . 3 |
11 | 10 | ex 114 | . 2 |
12 | 4 | adantr 274 | . . . 4 ↾t TopOn |
13 | cnprest.2 | . . . . . . . . 9 | |
14 | uniexg 4361 | . . . . . . . . 9 | |
15 | 13, 14 | eqeltrid 2226 | . . . . . . . 8 |
16 | 6, 15 | syl 14 | . . . . . . 7 |
17 | simprr 521 | . . . . . . 7 | |
18 | 16, 17 | ssexd 4068 | . . . . . 6 |
19 | resttop 12339 | . . . . . 6 ↾t | |
20 | 6, 18, 19 | syl2anc 408 | . . . . 5 ↾t |
21 | 20 | adantr 274 | . . . 4 ↾t ↾t |
22 | simpr 109 | . . . 4 ↾t ↾t | |
23 | cnprcl2k 12375 | . . . 4 TopOn ↾t ↾t | |
24 | 12, 21, 22, 23 | syl3anc 1216 | . . 3 ↾t |
25 | 24 | ex 114 | . 2 ↾t |
26 | simprl 520 | . . . . . . . . . 10 | |
27 | 26 | ffvelrnda 5555 | . . . . . . . . 9 |
28 | 27 | biantrud 302 | . . . . . . . 8 |
29 | elin 3259 | . . . . . . . 8 | |
30 | 28, 29 | syl6bbr 197 | . . . . . . 7 |
31 | imassrn 4892 | . . . . . . . . . . . 12 | |
32 | simplrl 524 | . . . . . . . . . . . . 13 | |
33 | 32 | frnd 5282 | . . . . . . . . . . . 12 |
34 | 31, 33 | sstrid 3108 | . . . . . . . . . . 11 |
35 | 34 | biantrud 302 | . . . . . . . . . 10 |
36 | ssin 3298 | . . . . . . . . . 10 | |
37 | 35, 36 | syl6bb 195 | . . . . . . . . 9 |
38 | 37 | anbi2d 459 | . . . . . . . 8 |
39 | 38 | rexbidv 2438 | . . . . . . 7 |
40 | 30, 39 | imbi12d 233 | . . . . . 6 |
41 | 40 | ralbidv 2437 | . . . . 5 |
42 | vex 2689 | . . . . . . . 8 | |
43 | 42 | inex1 4062 | . . . . . . 7 |
44 | 43 | a1i 9 | . . . . . 6 |
45 | 6 | adantr 274 | . . . . . . 7 |
46 | 18 | adantr 274 | . . . . . . 7 |
47 | elrest 12127 | . . . . . . 7 ↾t | |
48 | 45, 46, 47 | syl2anc 408 | . . . . . 6 ↾t |
49 | eleq2 2203 | . . . . . . . 8 | |
50 | sseq2 3121 | . . . . . . . . . 10 | |
51 | 50 | anbi2d 459 | . . . . . . . . 9 |
52 | 51 | rexbidv 2438 | . . . . . . . 8 |
53 | 49, 52 | imbi12d 233 | . . . . . . 7 |
54 | 53 | adantl 275 | . . . . . 6 |
55 | 44, 48, 54 | ralxfr2d 4385 | . . . . 5 ↾t |
56 | 41, 55 | bitr4d 190 | . . . 4 ↾t |
57 | 4 | adantr 274 | . . . . . 6 TopOn |
58 | 13 | toptopon 12185 | . . . . . . 7 TopOn |
59 | 45, 58 | sylib 121 | . . . . . 6 TopOn |
60 | simpr 109 | . . . . . 6 | |
61 | iscnp 12368 | . . . . . 6 TopOn TopOn | |
62 | 57, 59, 60, 61 | syl3anc 1216 | . . . . 5 |
63 | 17 | adantr 274 | . . . . . . 7 |
64 | 32, 63 | fssd 5285 | . . . . . 6 |
65 | 64 | biantrurd 303 | . . . . 5 |
66 | 62, 65 | bitr4d 190 | . . . 4 |
67 | resttopon 12340 | . . . . . . 7 TopOn ↾t TopOn | |
68 | 59, 63, 67 | syl2anc 408 | . . . . . 6 ↾t TopOn |
69 | iscnp 12368 | . . . . . 6 TopOn ↾t TopOn ↾t ↾t | |
70 | 57, 68, 60, 69 | syl3anc 1216 | . . . . 5 ↾t ↾t |
71 | 26 | biantrurd 303 | . . . . . 6 ↾t ↾t |
72 | 71 | adantr 274 | . . . . 5 ↾t ↾t |
73 | 70, 72 | bitr4d 190 | . . . 4 ↾t ↾t |
74 | 56, 66, 73 | 3bitr4d 219 | . . 3 ↾t |
75 | 74 | ex 114 | . 2 ↾t |
76 | 11, 25, 75 | pm5.21ndd 694 | 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 cuni 3736 crn 4540 cima 4542 wf 5119 cfv 5123 (class class class)co 5774 ↾t crest 12120 ctop 12164 TopOnctopon 12177 ccnp 12355 |
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-map 6544 df-rest 12122 df-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cnp 12358 |
This theorem is referenced by: (None) |
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