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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 12563 | . . . . . . 7 TopOn |
3 | 2 | biimpi 119 | . . . . . 6 TopOn |
4 | 3 | ad2antrr 480 | . . . . 5 TopOn |
5 | 4 | adantr 274 | . . . 4 TopOn |
6 | simplr 520 | . . . . 5 | |
7 | 6 | adantr 274 | . . . 4 |
8 | simpr 109 | . . . 4 | |
9 | cnprcl2k 12753 | . . . 4 TopOn | |
10 | 5, 7, 8, 9 | syl3anc 1227 | . . 3 |
11 | 10 | ex 114 | . 2 |
12 | 4 | adantr 274 | . . . 4 ↾t TopOn |
13 | cnprest.2 | . . . . . . . . 9 | |
14 | uniexg 4411 | . . . . . . . . 9 | |
15 | 13, 14 | eqeltrid 2251 | . . . . . . . 8 |
16 | 6, 15 | syl 14 | . . . . . . 7 |
17 | simprr 522 | . . . . . . 7 | |
18 | 16, 17 | ssexd 4116 | . . . . . 6 |
19 | resttop 12717 | . . . . . 6 ↾t | |
20 | 6, 18, 19 | syl2anc 409 | . . . . 5 ↾t |
21 | 20 | adantr 274 | . . . 4 ↾t ↾t |
22 | simpr 109 | . . . 4 ↾t ↾t | |
23 | cnprcl2k 12753 | . . . 4 TopOn ↾t ↾t | |
24 | 12, 21, 22, 23 | syl3anc 1227 | . . 3 ↾t |
25 | 24 | ex 114 | . 2 ↾t |
26 | simprl 521 | . . . . . . . . . 10 | |
27 | 26 | ffvelrnda 5614 | . . . . . . . . 9 |
28 | 27 | biantrud 302 | . . . . . . . 8 |
29 | elin 3300 | . . . . . . . 8 | |
30 | 28, 29 | bitr4di 197 | . . . . . . 7 |
31 | imassrn 4951 | . . . . . . . . . . . 12 | |
32 | simplrl 525 | . . . . . . . . . . . . 13 | |
33 | 32 | frnd 5341 | . . . . . . . . . . . 12 |
34 | 31, 33 | sstrid 3148 | . . . . . . . . . . 11 |
35 | 34 | biantrud 302 | . . . . . . . . . 10 |
36 | ssin 3339 | . . . . . . . . . 10 | |
37 | 35, 36 | bitrdi 195 | . . . . . . . . 9 |
38 | 37 | anbi2d 460 | . . . . . . . 8 |
39 | 38 | rexbidv 2465 | . . . . . . 7 |
40 | 30, 39 | imbi12d 233 | . . . . . 6 |
41 | 40 | ralbidv 2464 | . . . . 5 |
42 | vex 2724 | . . . . . . . 8 | |
43 | 42 | inex1 4110 | . . . . . . 7 |
44 | 43 | a1i 9 | . . . . . 6 |
45 | 6 | adantr 274 | . . . . . . 7 |
46 | 18 | adantr 274 | . . . . . . 7 |
47 | elrest 12505 | . . . . . . 7 ↾t | |
48 | 45, 46, 47 | syl2anc 409 | . . . . . 6 ↾t |
49 | eleq2 2228 | . . . . . . . 8 | |
50 | sseq2 3161 | . . . . . . . . . 10 | |
51 | 50 | anbi2d 460 | . . . . . . . . 9 |
52 | 51 | rexbidv 2465 | . . . . . . . 8 |
53 | 49, 52 | imbi12d 233 | . . . . . . 7 |
54 | 53 | adantl 275 | . . . . . 6 |
55 | 44, 48, 54 | ralxfr2d 4436 | . . . . 5 ↾t |
56 | 41, 55 | bitr4d 190 | . . . 4 ↾t |
57 | 4 | adantr 274 | . . . . . 6 TopOn |
58 | 13 | toptopon 12563 | . . . . . . 7 TopOn |
59 | 45, 58 | sylib 121 | . . . . . 6 TopOn |
60 | simpr 109 | . . . . . 6 | |
61 | iscnp 12746 | . . . . . 6 TopOn TopOn | |
62 | 57, 59, 60, 61 | syl3anc 1227 | . . . . 5 |
63 | 17 | adantr 274 | . . . . . . 7 |
64 | 32, 63 | fssd 5344 | . . . . . 6 |
65 | 64 | biantrurd 303 | . . . . 5 |
66 | 62, 65 | bitr4d 190 | . . . 4 |
67 | resttopon 12718 | . . . . . . 7 TopOn ↾t TopOn | |
68 | 59, 63, 67 | syl2anc 409 | . . . . . 6 ↾t TopOn |
69 | iscnp 12746 | . . . . . 6 TopOn ↾t TopOn ↾t ↾t | |
70 | 57, 68, 60, 69 | syl3anc 1227 | . . . . 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 695 | 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 cuni 3783 crn 4599 cima 4601 wf 5178 cfv 5182 (class class class)co 5836 ↾t crest 12498 ctop 12542 TopOnctopon 12555 ccnp 12733 |
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-map 6607 df-rest 12500 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cnp 12736 |
This theorem is referenced by: (None) |
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