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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 13087 | . . . . . . 7 TopOn |
3 | 2 | biimpi 120 | . . . . . 6 TopOn |
4 | 3 | ad2antrr 488 | . . . . 5 TopOn |
5 | 4 | adantr 276 | . . . 4 TopOn |
6 | simplr 528 | . . . . 5 | |
7 | 6 | adantr 276 | . . . 4 |
8 | simpr 110 | . . . 4 | |
9 | cnprcl2k 13277 | . . . 4 TopOn | |
10 | 5, 7, 8, 9 | syl3anc 1238 | . . 3 |
11 | 10 | ex 115 | . 2 |
12 | 4 | adantr 276 | . . . 4 ↾t TopOn |
13 | cnprest.2 | . . . . . . . . 9 | |
14 | uniexg 4433 | . . . . . . . . 9 | |
15 | 13, 14 | eqeltrid 2262 | . . . . . . . 8 |
16 | 6, 15 | syl 14 | . . . . . . 7 |
17 | simprr 531 | . . . . . . 7 | |
18 | 16, 17 | ssexd 4138 | . . . . . 6 |
19 | resttop 13241 | . . . . . 6 ↾t | |
20 | 6, 18, 19 | syl2anc 411 | . . . . 5 ↾t |
21 | 20 | adantr 276 | . . . 4 ↾t ↾t |
22 | simpr 110 | . . . 4 ↾t ↾t | |
23 | cnprcl2k 13277 | . . . 4 TopOn ↾t ↾t | |
24 | 12, 21, 22, 23 | syl3anc 1238 | . . 3 ↾t |
25 | 24 | ex 115 | . 2 ↾t |
26 | simprl 529 | . . . . . . . . . 10 | |
27 | 26 | ffvelcdmda 5643 | . . . . . . . . 9 |
28 | 27 | biantrud 304 | . . . . . . . 8 |
29 | elin 3316 | . . . . . . . 8 | |
30 | 28, 29 | bitr4di 198 | . . . . . . 7 |
31 | imassrn 4974 | . . . . . . . . . . . 12 | |
32 | simplrl 535 | . . . . . . . . . . . . 13 | |
33 | 32 | frnd 5367 | . . . . . . . . . . . 12 |
34 | 31, 33 | sstrid 3164 | . . . . . . . . . . 11 |
35 | 34 | biantrud 304 | . . . . . . . . . 10 |
36 | ssin 3355 | . . . . . . . . . 10 | |
37 | 35, 36 | bitrdi 196 | . . . . . . . . 9 |
38 | 37 | anbi2d 464 | . . . . . . . 8 |
39 | 38 | rexbidv 2476 | . . . . . . 7 |
40 | 30, 39 | imbi12d 234 | . . . . . 6 |
41 | 40 | ralbidv 2475 | . . . . 5 |
42 | vex 2738 | . . . . . . . 8 | |
43 | 42 | inex1 4132 | . . . . . . 7 |
44 | 43 | a1i 9 | . . . . . 6 |
45 | 6 | adantr 276 | . . . . . . 7 |
46 | 18 | adantr 276 | . . . . . . 7 |
47 | elrest 12617 | . . . . . . 7 ↾t | |
48 | 45, 46, 47 | syl2anc 411 | . . . . . 6 ↾t |
49 | eleq2 2239 | . . . . . . . 8 | |
50 | sseq2 3177 | . . . . . . . . . 10 | |
51 | 50 | anbi2d 464 | . . . . . . . . 9 |
52 | 51 | rexbidv 2476 | . . . . . . . 8 |
53 | 49, 52 | imbi12d 234 | . . . . . . 7 |
54 | 53 | adantl 277 | . . . . . 6 |
55 | 44, 48, 54 | ralxfr2d 4458 | . . . . 5 ↾t |
56 | 41, 55 | bitr4d 191 | . . . 4 ↾t |
57 | 4 | adantr 276 | . . . . . 6 TopOn |
58 | 13 | toptopon 13087 | . . . . . . 7 TopOn |
59 | 45, 58 | sylib 122 | . . . . . 6 TopOn |
60 | simpr 110 | . . . . . 6 | |
61 | iscnp 13270 | . . . . . 6 TopOn TopOn | |
62 | 57, 59, 60, 61 | syl3anc 1238 | . . . . 5 |
63 | 17 | adantr 276 | . . . . . . 7 |
64 | 32, 63 | fssd 5370 | . . . . . 6 |
65 | 64 | biantrurd 305 | . . . . 5 |
66 | 62, 65 | bitr4d 191 | . . . 4 |
67 | resttopon 13242 | . . . . . . 7 TopOn ↾t TopOn | |
68 | 59, 63, 67 | syl2anc 411 | . . . . . 6 ↾t TopOn |
69 | iscnp 13270 | . . . . . 6 TopOn ↾t TopOn ↾t ↾t | |
70 | 57, 68, 60, 69 | syl3anc 1238 | . . . . 5 ↾t ↾t |
71 | 26 | biantrurd 305 | . . . . . 6 ↾t ↾t |
72 | 71 | adantr 276 | . . . . 5 ↾t ↾t |
73 | 70, 72 | bitr4d 191 | . . . 4 ↾t ↾t |
74 | 56, 66, 73 | 3bitr4d 220 | . . 3 ↾t |
75 | 74 | ex 115 | . 2 ↾t |
76 | 11, 25, 75 | pm5.21ndd 705 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 wrex 2454 cvv 2735 cin 3126 wss 3127 cuni 3805 crn 4621 cima 4623 wf 5204 cfv 5208 (class class class)co 5865 ↾t crest 12610 ctop 13066 TopOnctopon 13079 ccnp 13257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-map 6640 df-rest 12612 df-topgen 12631 df-top 13067 df-topon 13080 df-bases 13112 df-cnp 13260 |
This theorem is referenced by: (None) |
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