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Mirrors > Home > ILE Home > Th. List > frecuzrdgdomlem | Unicode version |
Description: The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c |
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frecuzrdgrclt.a |
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frecuzrdgrclt.t |
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frecuzrdgrclt.f |
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frecuzrdgrclt.r |
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frecuzrdgdomlem.g |
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Ref | Expression |
---|---|
frecuzrdgdomlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c |
. . . . . 6
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2 | frecuzrdgrclt.a |
. . . . . 6
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3 | frecuzrdgrclt.t |
. . . . . 6
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4 | frecuzrdgrclt.f |
. . . . . 6
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5 | frecuzrdgrclt.r |
. . . . . 6
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6 | 1, 2, 3, 4, 5 | frecuzrdgrclt 10081 |
. . . . 5
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7 | frn 5239 |
. . . . 5
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8 | 6, 7 | syl 14 |
. . . 4
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9 | dmss 4698 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | dmxpss 4927 |
. . 3
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12 | 10, 11 | syl6ss 3075 |
. 2
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13 | 8 | adantr 272 |
. . . . . . . . 9
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14 | ffun 5233 |
. . . . . . . . . . . 12
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15 | 6, 14 | syl 14 |
. . . . . . . . . . 11
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16 | 15 | adantr 272 |
. . . . . . . . . 10
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17 | frecuzrdgdomlem.g |
. . . . . . . . . . . . 13
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18 | 1, 17 | frec2uzf1od 10072 |
. . . . . . . . . . . 12
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19 | f1ocnvdm 5636 |
. . . . . . . . . . . 12
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20 | 18, 19 | sylan 279 |
. . . . . . . . . . 11
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21 | fdm 5236 |
. . . . . . . . . . . . 13
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22 | 6, 21 | syl 14 |
. . . . . . . . . . . 12
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23 | 22 | adantr 272 |
. . . . . . . . . . 11
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24 | 20, 23 | eleqtrrd 2194 |
. . . . . . . . . 10
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25 | fvelrn 5505 |
. . . . . . . . . 10
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26 | 16, 24, 25 | syl2anc 406 |
. . . . . . . . 9
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27 | 13, 26 | sseldd 3064 |
. . . . . . . 8
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28 | 1st2nd2 6027 |
. . . . . . . 8
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29 | 27, 28 | syl 14 |
. . . . . . 7
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30 | 1 | adantr 272 |
. . . . . . . . . 10
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31 | 2 | adantr 272 |
. . . . . . . . . 10
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32 | 3 | adantr 272 |
. . . . . . . . . 10
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33 | 4 | adantlr 466 |
. . . . . . . . . 10
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34 | 30, 31, 32, 33, 5, 20, 17 | frecuzrdgg 10082 |
. . . . . . . . 9
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35 | f1ocnvfv2 5633 |
. . . . . . . . . 10
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36 | 18, 35 | sylan 279 |
. . . . . . . . 9
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37 | 34, 36 | eqtrd 2147 |
. . . . . . . 8
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38 | 37 | opeq1d 3677 |
. . . . . . 7
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39 | 29, 38 | eqtrd 2147 |
. . . . . 6
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40 | 39, 26 | eqeltrrd 2192 |
. . . . 5
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41 | simpr 109 |
. . . . . 6
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42 | xp2nd 6018 |
. . . . . . 7
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43 | 27, 42 | syl 14 |
. . . . . 6
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44 | opeldmg 4704 |
. . . . . 6
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45 | 41, 43, 44 | syl2anc 406 |
. . . . 5
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46 | 40, 45 | mpd 13 |
. . . 4
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47 | 46 | ex 114 |
. . 3
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48 | 47 | ssrdv 3069 |
. 2
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49 | 12, 48 | eqssd 3080 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-ilim 4251 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-frec 6242 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-inn 8631 df-n0 8882 df-z 8959 df-uz 9229 |
This theorem is referenced by: frecuzrdgdom 10084 |
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