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Mirrors > Home > ILE Home > Th. List > frec2uzrand | Unicode version |
Description: Range of (see frec2uz0d 10328). (Contributed by Jim Kingdon, 17-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
Ref | Expression |
---|---|
frec2uzrand |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . 2 | |
2 | zex 9194 | . . . . . . . . . . 11 | |
3 | 2 | mptex 5708 | . . . . . . . . . 10 |
4 | vex 2727 | . . . . . . . . . 10 | |
5 | 3, 4 | fvex 5503 | . . . . . . . . 9 |
6 | 5 | ax-gen 1436 | . . . . . . . 8 |
7 | frecfnom 6363 | . . . . . . . 8 frec | |
8 | 6, 7 | mpan 421 | . . . . . . 7 frec |
9 | frec2uz.2 | . . . . . . . 8 frec | |
10 | 9 | fneq1i 5279 | . . . . . . 7 frec |
11 | 8, 10 | sylibr 133 | . . . . . 6 |
12 | fvelrnb 5531 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | simpl 108 | . . . . . . . 8 | |
15 | simpr 109 | . . . . . . . 8 | |
16 | 14, 9, 15 | frec2uzuzd 10331 | . . . . . . 7 |
17 | eleq1 2227 | . . . . . . 7 | |
18 | 16, 17 | syl5ibcom 154 | . . . . . 6 |
19 | 18 | rexlimdva 2581 | . . . . 5 |
20 | 13, 19 | sylbid 149 | . . . 4 |
21 | eleq1 2227 | . . . . 5 | |
22 | eleq1 2227 | . . . . 5 | |
23 | eleq1 2227 | . . . . 5 | |
24 | id 19 | . . . . . . 7 | |
25 | 24, 9 | frec2uz0d 10328 | . . . . . 6 |
26 | peano1 4568 | . . . . . . 7 | |
27 | fnfvelrn 5614 | . . . . . . 7 | |
28 | 11, 26, 27 | sylancl 410 | . . . . . 6 |
29 | 25, 28 | eqeltrrd 2242 | . . . . 5 |
30 | eluzel2 9465 | . . . . . 6 | |
31 | 14, 9, 15 | frec2uzsucd 10330 | . . . . . . . . . . 11 |
32 | oveq1 5846 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan9eq 2217 | . . . . . . . . . 10 |
34 | peano2 4569 | . . . . . . . . . . . 12 | |
35 | fnfvelrn 5614 | . . . . . . . . . . . 12 | |
36 | 11, 34, 35 | syl2an 287 | . . . . . . . . . . 11 |
37 | 36 | adantr 274 | . . . . . . . . . 10 |
38 | 33, 37 | eqeltrrd 2242 | . . . . . . . . 9 |
39 | 38 | ex 114 | . . . . . . . 8 |
40 | 39 | rexlimdva 2581 | . . . . . . 7 |
41 | 13, 40 | sylbid 149 | . . . . . 6 |
42 | 30, 41 | syl 14 | . . . . 5 |
43 | 21, 22, 23, 22, 29, 42 | uzind4 9520 | . . . 4 |
44 | 20, 43 | impbid1 141 | . . 3 |
45 | 44 | eqrdv 2162 | . 2 |
46 | 1, 45 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1340 wceq 1342 wcel 2135 wrex 2443 cvv 2724 c0 3407 cmpt 4040 csuc 4340 com 4564 crn 4602 wfn 5180 cfv 5185 (class class class)co 5839 freccfrec 6352 c1 7748 caddc 7750 cz 9185 cuz 9460 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-addcom 7847 ax-addass 7849 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-0id 7855 ax-rnegex 7856 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-ltadd 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-recs 6267 df-frec 6353 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-inn 8852 df-n0 9109 df-z 9186 df-uz 9461 |
This theorem is referenced by: frec2uzf1od 10335 |
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