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Mirrors > Home > ILE Home > Th. List > frec2uzf1od | Unicode version |
Description: (see frec2uz0d 10172) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
Ref | Expression |
---|---|
frec2uzf1od |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 9063 | . . . . . . . . 9 | |
2 | 1 | mptex 5646 | . . . . . . . 8 |
3 | vex 2689 | . . . . . . . 8 | |
4 | 2, 3 | fvex 5441 | . . . . . . 7 |
5 | 4 | ax-gen 1425 | . . . . . 6 |
6 | frec2uz.1 | . . . . . 6 | |
7 | frecfnom 6298 | . . . . . 6 frec | |
8 | 5, 6, 7 | sylancr 410 | . . . . 5 frec |
9 | frec2uz.2 | . . . . . 6 frec | |
10 | 9 | fneq1i 5217 | . . . . 5 frec |
11 | 8, 10 | sylibr 133 | . . . 4 |
12 | 6, 9 | frec2uzrand 10178 | . . . . 5 |
13 | eqimss 3151 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | df-f 5127 | . . . 4 | |
16 | 11, 14, 15 | sylanbrc 413 | . . 3 |
17 | 6 | adantr 274 | . . . . . . . . . . . . . 14 |
18 | simpr 109 | . . . . . . . . . . . . . 14 | |
19 | 17, 9, 18 | frec2uzzd 10173 | . . . . . . . . . . . . 13 |
20 | 19 | 3adant3 1001 | . . . . . . . . . . . 12 |
21 | 20 | zred 9173 | . . . . . . . . . . 11 |
22 | 21 | ltnrd 7875 | . . . . . . . . . 10 |
23 | 22 | adantr 274 | . . . . . . . . 9 |
24 | simpr 109 | . . . . . . . . . 10 | |
25 | 24 | breq2d 3941 | . . . . . . . . 9 |
26 | 23, 25 | mtbid 661 | . . . . . . . 8 |
27 | 17 | 3adant3 1001 | . . . . . . . . . . 11 |
28 | simp2 982 | . . . . . . . . . . 11 | |
29 | simp3 983 | . . . . . . . . . . 11 | |
30 | 27, 9, 28, 29 | frec2uzltd 10176 | . . . . . . . . . 10 |
31 | 30 | con3d 620 | . . . . . . . . 9 |
32 | 31 | adantr 274 | . . . . . . . 8 |
33 | 26, 32 | mpd 13 | . . . . . . 7 |
34 | 24 | breq1d 3939 | . . . . . . . . 9 |
35 | 23, 34 | mtbid 661 | . . . . . . . 8 |
36 | 27, 9, 29, 28 | frec2uzltd 10176 | . . . . . . . . 9 |
37 | 36 | adantr 274 | . . . . . . . 8 |
38 | 35, 37 | mtod 652 | . . . . . . 7 |
39 | nntri3 6393 | . . . . . . . . 9 | |
40 | 39 | 3adant1 999 | . . . . . . . 8 |
41 | 40 | adantr 274 | . . . . . . 7 |
42 | 33, 38, 41 | mpbir2and 928 | . . . . . 6 |
43 | 42 | ex 114 | . . . . 5 |
44 | 43 | 3expb 1182 | . . . 4 |
45 | 44 | ralrimivva 2514 | . . 3 |
46 | dff13 5669 | . . 3 | |
47 | 16, 45, 46 | sylanbrc 413 | . 2 |
48 | dff1o5 5376 | . 2 | |
49 | 47, 12, 48 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2416 cvv 2686 wss 3071 class class class wbr 3929 cmpt 3989 com 4504 crn 4540 wfn 5118 wf 5119 wf1 5120 wf1o 5122 cfv 5123 (class class class)co 5774 freccfrec 6287 c1 7621 caddc 7623 clt 7800 cz 9054 cuz 9326 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: frec2uzisod 10180 frecuzrdglem 10184 frecuzrdgtcl 10185 frecuzrdgsuc 10187 frecuzrdgg 10189 frecuzrdgdomlem 10190 frecuzrdgfunlem 10192 frecuzrdgsuctlem 10196 uzenom 10198 frecfzennn 10199 frechashgf1o 10201 frec2uzled 10202 hashfz1 10529 hashen 10530 ennnfonelemjn 11915 ennnfonelem1 11920 ennnfonelemhf1o 11926 ennnfonelemrn 11932 |
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