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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftmo | Structured version Visualization version Unicode version |
Description: A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
cvmliftmo.b | |
cvmliftmo.y | |
cvmliftmo.f | CovMap |
cvmliftmo.k | Conn |
cvmliftmo.l | 𝑛Locally Conn |
cvmliftmo.o | |
cvmliftmo.g | |
cvmliftmo.p | |
cvmliftmo.e |
Ref | Expression |
---|---|
cvmliftmo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftmo.b | . . . . 5 | |
2 | cvmliftmo.y | . . . . 5 | |
3 | cvmliftmo.f | . . . . . 6 CovMap | |
4 | 3 | ad2antrr 762 | . . . . 5 CovMap |
5 | cvmliftmo.k | . . . . . 6 Conn | |
6 | 5 | ad2antrr 762 | . . . . 5 Conn |
7 | cvmliftmo.l | . . . . . 6 𝑛Locally Conn | |
8 | 7 | ad2antrr 762 | . . . . 5 𝑛Locally Conn |
9 | cvmliftmo.o | . . . . . 6 | |
10 | 9 | ad2antrr 762 | . . . . 5 |
11 | simplrl 800 | . . . . 5 | |
12 | simplrr 801 | . . . . 5 | |
13 | simprll 802 | . . . . . 6 | |
14 | simprrl 804 | . . . . . 6 | |
15 | 13, 14 | eqtr4d 2659 | . . . . 5 |
16 | simprlr 803 | . . . . . 6 | |
17 | simprrr 805 | . . . . . 6 | |
18 | 16, 17 | eqtr4d 2659 | . . . . 5 |
19 | 1, 2, 4, 6, 8, 10, 11, 12, 15, 18 | cvmliftmoi 31265 | . . . 4 |
20 | 19 | ex 450 | . . 3 |
21 | 20 | ralrimivva 2971 | . 2 |
22 | coeq2 5280 | . . . . 5 | |
23 | 22 | eqeq1d 2624 | . . . 4 |
24 | fveq1 6190 | . . . . 5 | |
25 | 24 | eqeq1d 2624 | . . . 4 |
26 | 23, 25 | anbi12d 747 | . . 3 |
27 | 26 | rmo4 3399 | . 2 |
28 | 21, 27 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wrmo 2915 cuni 4436 ccom 5118 cfv 5888 (class class class)co 6650 ccn 21028 Conncconn 21214 𝑛Locally cnlly 21268 CovMap ccvm 31237 |
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-rmo 2920 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-riota 6611 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-cld 20823 df-nei 20902 df-cn 21031 df-conn 21215 df-nlly 21270 df-hmeo 21558 df-cvm 31238 |
This theorem is referenced by: cvmliftlem14 31279 cvmlift2lem13 31297 cvmlift3 31310 |
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