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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 10428 |
. . . . 5
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7 | frn 5386 |
. . . . 5
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8 | 6, 7 | syl 14 |
. . . 4
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9 | dmss 4838 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | dmxpss 5071 |
. . 3
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12 | 10, 11 | sstrdi 3179 |
. 2
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13 | 8 | adantr 276 |
. . . . . . . . 9
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14 | ffun 5380 |
. . . . . . . . . . . 12
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15 | 6, 14 | syl 14 |
. . . . . . . . . . 11
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16 | 15 | adantr 276 |
. . . . . . . . . 10
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17 | frecuzrdgdomlem.g |
. . . . . . . . . . . . 13
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18 | 1, 17 | frec2uzf1od 10419 |
. . . . . . . . . . . 12
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19 | f1ocnvdm 5795 |
. . . . . . . . . . . 12
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20 | 18, 19 | sylan 283 |
. . . . . . . . . . 11
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21 | fdm 5383 |
. . . . . . . . . . . . 13
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22 | 6, 21 | syl 14 |
. . . . . . . . . . . 12
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23 | 22 | adantr 276 |
. . . . . . . . . . 11
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24 | 20, 23 | eleqtrrd 2267 |
. . . . . . . . . 10
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25 | fvelrn 5660 |
. . . . . . . . . 10
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26 | 16, 24, 25 | syl2anc 411 |
. . . . . . . . 9
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27 | 13, 26 | sseldd 3168 |
. . . . . . . 8
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28 | 1st2nd2 6189 |
. . . . . . . 8
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29 | 27, 28 | syl 14 |
. . . . . . 7
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30 | 1 | adantr 276 |
. . . . . . . . . 10
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31 | 2 | adantr 276 |
. . . . . . . . . 10
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32 | 3 | adantr 276 |
. . . . . . . . . 10
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33 | 4 | adantlr 477 |
. . . . . . . . . 10
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34 | 30, 31, 32, 33, 5, 20, 17 | frecuzrdgg 10429 |
. . . . . . . . 9
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35 | f1ocnvfv2 5792 |
. . . . . . . . . 10
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36 | 18, 35 | sylan 283 |
. . . . . . . . 9
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37 | 34, 36 | eqtrd 2220 |
. . . . . . . 8
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38 | 37 | opeq1d 3796 |
. . . . . . 7
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39 | 29, 38 | eqtrd 2220 |
. . . . . 6
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40 | 39, 26 | eqeltrrd 2265 |
. . . . 5
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41 | simpr 110 |
. . . . . 6
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42 | xp2nd 6180 |
. . . . . . 7
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43 | 27, 42 | syl 14 |
. . . . . 6
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44 | opeldmg 4844 |
. . . . . 6
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45 | 41, 43, 44 | syl2anc 411 |
. . . . 5
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46 | 40, 45 | mpd 13 |
. . . 4
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47 | 46 | ex 115 |
. . 3
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48 | 47 | ssrdv 3173 |
. 2
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49 | 12, 48 | eqssd 3184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 |
This theorem is referenced by: frecuzrdgdom 10431 |
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