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Mirrors > Home > MPE Home > Th. List > cnrest2 | Structured version Visualization version Unicode version |
Description: Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnrest2 | TopOn ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 21044 | . . . 4 | |
2 | 1 | a1i 11 | . . 3 TopOn |
3 | eqid 2622 | . . . . . . . 8 | |
4 | eqid 2622 | . . . . . . . 8 | |
5 | 3, 4 | cnf 21050 | . . . . . . 7 |
6 | ffn 6045 | . . . . . . 7 | |
7 | 5, 6 | syl 17 | . . . . . 6 |
8 | 7 | a1i 11 | . . . . 5 TopOn |
9 | simp2 1062 | . . . . 5 TopOn | |
10 | 8, 9 | jctird 567 | . . . 4 TopOn |
11 | df-f 5892 | . . . 4 | |
12 | 10, 11 | syl6ibr 242 | . . 3 TopOn |
13 | 2, 12 | jcad 555 | . 2 TopOn |
14 | cntop1 21044 | . . . . 5 ↾t | |
15 | 14 | adantl 482 | . . . 4 TopOn ↾t |
16 | 3 | toptopon 20722 | . . . . . 6 TopOn |
17 | 15, 16 | sylib 208 | . . . . 5 TopOn ↾t TopOn |
18 | resttopon 20965 | . . . . . . 7 TopOn ↾t TopOn | |
19 | 18 | 3adant2 1080 | . . . . . 6 TopOn ↾t TopOn |
20 | 19 | adantr 481 | . . . . 5 TopOn ↾t ↾t TopOn |
21 | simpr 477 | . . . . 5 TopOn ↾t ↾t | |
22 | cnf2 21053 | . . . . 5 TopOn ↾t TopOn ↾t | |
23 | 17, 20, 21, 22 | syl3anc 1326 | . . . 4 TopOn ↾t |
24 | 15, 23 | jca 554 | . . 3 TopOn ↾t |
25 | 24 | ex 450 | . 2 TopOn ↾t |
26 | vex 3203 | . . . . . . . . 9 | |
27 | 26 | inex1 4799 | . . . . . . . 8 |
28 | 27 | a1i 11 | . . . . . . 7 TopOn |
29 | simpl1 1064 | . . . . . . . 8 TopOn TopOn | |
30 | toponmax 20730 | . . . . . . . . . 10 TopOn | |
31 | 29, 30 | syl 17 | . . . . . . . . 9 TopOn |
32 | simpl3 1066 | . . . . . . . . 9 TopOn | |
33 | 31, 32 | ssexd 4805 | . . . . . . . 8 TopOn |
34 | elrest 16088 | . . . . . . . 8 TopOn ↾t | |
35 | 29, 33, 34 | syl2anc 693 | . . . . . . 7 TopOn ↾t |
36 | imaeq2 5462 | . . . . . . . . 9 | |
37 | 36 | eleq1d 2686 | . . . . . . . 8 |
38 | 37 | adantl 482 | . . . . . . 7 TopOn |
39 | 28, 35, 38 | ralxfr2d 4882 | . . . . . 6 TopOn ↾t |
40 | simplrr 801 | . . . . . . . . . 10 TopOn | |
41 | ffun 6048 | . . . . . . . . . 10 | |
42 | inpreima 6342 | . . . . . . . . . 10 | |
43 | 40, 41, 42 | 3syl 18 | . . . . . . . . 9 TopOn |
44 | cnvimass 5485 | . . . . . . . . . . . 12 | |
45 | cnvimarndm 5486 | . . . . . . . . . . . 12 | |
46 | 44, 45 | sseqtr4i 3638 | . . . . . . . . . . 11 |
47 | simpll2 1101 | . . . . . . . . . . . 12 TopOn | |
48 | imass2 5501 | . . . . . . . . . . . 12 | |
49 | 47, 48 | syl 17 | . . . . . . . . . . 11 TopOn |
50 | 46, 49 | syl5ss 3614 | . . . . . . . . . 10 TopOn |
51 | df-ss 3588 | . . . . . . . . . 10 | |
52 | 50, 51 | sylib 208 | . . . . . . . . 9 TopOn |
53 | 43, 52 | eqtrd 2656 | . . . . . . . 8 TopOn |
54 | 53 | eleq1d 2686 | . . . . . . 7 TopOn |
55 | 54 | ralbidva 2985 | . . . . . 6 TopOn |
56 | simprr 796 | . . . . . . . 8 TopOn | |
57 | 56, 32 | fssd 6057 | . . . . . . 7 TopOn |
58 | 57 | biantrurd 529 | . . . . . 6 TopOn |
59 | 39, 55, 58 | 3bitrrd 295 | . . . . 5 TopOn ↾t |
60 | 56 | biantrurd 529 | . . . . 5 TopOn ↾t ↾t |
61 | 59, 60 | bitrd 268 | . . . 4 TopOn ↾t |
62 | simprl 794 | . . . . . 6 TopOn | |
63 | 62, 16 | sylib 208 | . . . . 5 TopOn TopOn |
64 | iscn 21039 | . . . . 5 TopOn TopOn | |
65 | 63, 29, 64 | syl2anc 693 | . . . 4 TopOn |
66 | 19 | adantr 481 | . . . . 5 TopOn ↾t TopOn |
67 | iscn 21039 | . . . . 5 TopOn ↾t TopOn ↾t ↾t | |
68 | 63, 66, 67 | syl2anc 693 | . . . 4 TopOn ↾t ↾t |
69 | 61, 65, 68 | 3bitr4d 300 | . . 3 TopOn ↾t |
70 | 69 | ex 450 | . 2 TopOn ↾t |
71 | 13, 25, 70 | pm5.21ndd 369 | 1 TopOn ↾t |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 cvv 3200 cin 3573 wss 3574 cuni 4436 ccnv 5113 cdm 5114 crn 5115 cima 5117 wfun 5882 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650 ↾t crest 16081 ctop 20698 TopOnctopon 20715 ccn 21028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 |
This theorem is referenced by: cnrest2r 21091 rncmp 21199 connima 21228 conncn 21229 kgencn2 21360 kgencn3 21361 qtoprest 21520 hmeores 21574 symgtgp 21905 submtmd 21908 subgtgp 21909 metdcn2 22642 metdscn2 22660 cnmptre 22726 iimulcn 22737 icchmeo 22740 evth 22758 evth2 22759 lebnumlem2 22761 reparphti 22797 efrlim 24696 rmulccn 29974 raddcn 29975 xrge0mulc1cn 29987 cvxpconn 31224 cvxsconn 31225 cvmliftmolem1 31263 cvmliftlem8 31274 cvmlift2lem9 31293 cvmlift3lem6 31306 ivthALT 32330 knoppcnlem10 32492 broucube 33443 areacirclem2 33501 cnres2 33562 cnresima 33563 refsumcn 39189 icccncfext 40100 |
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