Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1ocnt | Structured version Visualization version Unicode version |
Description: Given a countable set , number its elements by providing a one-to-one mapping either with or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 29557 or iundisj2cnt 29558. (Contributed by Thierry Arnoux, 25-Jul-2020.) |
Ref | Expression |
---|---|
f1ocnt | ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1o0 6173 | . . . . . . 7 | |
2 | eqidd 2623 | . . . . . . . 8 | |
3 | dm0 5339 | . . . . . . . . 9 | |
4 | 3 | a1i 11 | . . . . . . . 8 |
5 | id 22 | . . . . . . . 8 | |
6 | 2, 4, 5 | f1oeq123d 6133 | . . . . . . 7 |
7 | 1, 6 | mpbiri 248 | . . . . . 6 |
8 | fveq2 6191 | . . . . . . . . . . . . 13 | |
9 | hash0 13158 | . . . . . . . . . . . . 13 | |
10 | 8, 9 | syl6eq 2672 | . . . . . . . . . . . 12 |
11 | 10 | oveq1d 6665 | . . . . . . . . . . 11 |
12 | 0p1e1 11132 | . . . . . . . . . . 11 | |
13 | 11, 12 | syl6eq 2672 | . . . . . . . . . 10 |
14 | 13 | oveq2d 6666 | . . . . . . . . 9 ..^ ..^ |
15 | fzo0 12492 | . . . . . . . . 9 ..^ | |
16 | 14, 15 | syl6eq 2672 | . . . . . . . 8 ..^ |
17 | 4, 16 | eqtr4d 2659 | . . . . . . 7 ..^ |
18 | 17 | olcd 408 | . . . . . 6 ..^ |
19 | 7, 18 | jca 554 | . . . . 5 ..^ |
20 | 0ex 4790 | . . . . . 6 | |
21 | id 22 | . . . . . . . 8 | |
22 | dmeq 5324 | . . . . . . . 8 | |
23 | eqidd 2623 | . . . . . . . 8 | |
24 | 21, 22, 23 | f1oeq123d 6133 | . . . . . . 7 |
25 | 22 | eqeq1d 2624 | . . . . . . . 8 |
26 | 22 | eqeq1d 2624 | . . . . . . . 8 ..^ ..^ |
27 | 25, 26 | orbi12d 746 | . . . . . . 7 ..^ ..^ |
28 | 24, 27 | anbi12d 747 | . . . . . 6 ..^ ..^ |
29 | 20, 28 | spcev 3300 | . . . . 5 ..^ ..^ |
30 | 19, 29 | syl 17 | . . . 4 ..^ |
31 | 30 | adantl 482 | . . 3 ..^ |
32 | f1odm 6141 | . . . . . . . . . . 11 | |
33 | f1oeq2 6128 | . . . . . . . . . . 11 | |
34 | 32, 33 | syl 17 | . . . . . . . . . 10 |
35 | 34 | ibir 257 | . . . . . . . . 9 |
36 | 35 | adantl 482 | . . . . . . . 8 |
37 | 32 | adantl 482 | . . . . . . . . . 10 |
38 | simpl 473 | . . . . . . . . . . . 12 | |
39 | 38 | nnzd 11481 | . . . . . . . . . . 11 |
40 | fzval3 12536 | . . . . . . . . . . 11 ..^ | |
41 | 39, 40 | syl 17 | . . . . . . . . . 10 ..^ |
42 | 37, 41 | eqtrd 2656 | . . . . . . . . 9 ..^ |
43 | 42 | olcd 408 | . . . . . . . 8 ..^ |
44 | 36, 43 | jca 554 | . . . . . . 7 ..^ |
45 | 44 | ex 450 | . . . . . 6 ..^ |
46 | 45 | eximdv 1846 | . . . . 5 ..^ |
47 | 46 | imp 445 | . . . 4 ..^ |
48 | 47 | adantl 482 | . . 3 ..^ |
49 | fz1f1o 14441 | . . . 4 | |
50 | 49 | adantl 482 | . . 3 |
51 | 31, 48, 50 | mpjaodan 827 | . 2 ..^ |
52 | isfinite 8549 | . . . . . . . . . 10 | |
53 | 52 | notbii 310 | . . . . . . . . 9 |
54 | 53 | biimpi 206 | . . . . . . . 8 |
55 | 54 | anim2i 593 | . . . . . . 7 |
56 | bren2 7986 | . . . . . . 7 | |
57 | 55, 56 | sylibr 224 | . . . . . 6 |
58 | nnenom 12779 | . . . . . . 7 | |
59 | 58 | ensymi 8006 | . . . . . 6 |
60 | entr 8008 | . . . . . 6 | |
61 | 57, 59, 60 | sylancl 694 | . . . . 5 |
62 | bren 7964 | . . . . 5 | |
63 | 61, 62 | sylib 208 | . . . 4 |
64 | f1oexbi 7116 | . . . 4 | |
65 | 63, 64 | sylib 208 | . . 3 |
66 | f1odm 6141 | . . . . . . 7 | |
67 | f1oeq2 6128 | . . . . . . 7 | |
68 | 66, 67 | syl 17 | . . . . . 6 |
69 | 68 | ibir 257 | . . . . 5 |
70 | 66 | orcd 407 | . . . . 5 ..^ |
71 | 69, 70 | jca 554 | . . . 4 ..^ |
72 | 71 | eximi 1762 | . . 3 ..^ |
73 | 65, 72 | syl 17 | . 2 ..^ |
74 | 51, 73 | pm2.61dan 832 | 1 ..^ |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wex 1704 wcel 1990 c0 3915 class class class wbr 4653 cdm 5114 wf1o 5887 cfv 5888 (class class class)co 6650 com 7065 cen 7952 cdom 7953 csdm 7954 cfn 7955 cc0 9936 c1 9937 caddc 9939 cn 11020 cz 11377 cfz 12326 ..^cfzo 12465 chash 13117 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-nel 2898 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-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-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 |
This theorem is referenced by: (None) |
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