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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 9711 |
. . . . 5
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7 | frn 5121 |
. . . . 5
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8 | 6, 7 | syl 14 |
. . . 4
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9 | dmss 4593 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | dmxpss 4815 |
. . 3
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12 | 10, 11 | syl6ss 3022 |
. 2
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13 | 8 | adantr 270 |
. . . . . . . . 9
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14 | ffun 5117 |
. . . . . . . . . . . 12
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15 | 6, 14 | syl 14 |
. . . . . . . . . . 11
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16 | 15 | adantr 270 |
. . . . . . . . . 10
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17 | frecuzrdgdomlem.g |
. . . . . . . . . . . . 13
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18 | 1, 17 | frec2uzf1od 9702 |
. . . . . . . . . . . 12
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19 | f1ocnvdm 5500 |
. . . . . . . . . . . 12
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20 | 18, 19 | sylan 277 |
. . . . . . . . . . 11
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21 | fdm 5119 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 6, 21 | syl 14 |
. . . . . . . . . . . 12
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23 | 22 | adantr 270 |
. . . . . . . . . . 11
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24 | 20, 23 | eleqtrrd 2162 |
. . . . . . . . . 10
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25 | fvelrn 5375 |
. . . . . . . . . 10
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26 | 16, 24, 25 | syl2anc 403 |
. . . . . . . . 9
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27 | 13, 26 | sseldd 3011 |
. . . . . . . 8
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28 | 1st2nd2 5880 |
. . . . . . . 8
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29 | 27, 28 | syl 14 |
. . . . . . 7
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30 | 1 | adantr 270 |
. . . . . . . . . 10
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31 | 2 | adantr 270 |
. . . . . . . . . 10
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32 | 3 | adantr 270 |
. . . . . . . . . 10
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33 | 4 | adantlr 461 |
. . . . . . . . . 10
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34 | 30, 31, 32, 33, 5, 20, 17 | frecuzrdgg 9712 |
. . . . . . . . 9
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35 | f1ocnvfv2 5497 |
. . . . . . . . . 10
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36 | 18, 35 | sylan 277 |
. . . . . . . . 9
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37 | 34, 36 | eqtrd 2115 |
. . . . . . . 8
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38 | 37 | opeq1d 3602 |
. . . . . . 7
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39 | 29, 38 | eqtrd 2115 |
. . . . . 6
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40 | 39, 26 | eqeltrrd 2160 |
. . . . 5
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41 | simpr 108 |
. . . . . 6
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42 | xp2nd 5872 |
. . . . . . 7
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43 | 27, 42 | syl 14 |
. . . . . 6
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44 | opeldmg 4599 |
. . . . . 6
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45 | 41, 43, 44 | syl2anc 403 |
. . . . 5
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46 | 40, 45 | mpd 13 |
. . . 4
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47 | 46 | ex 113 |
. . 3
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48 | 47 | ssrdv 3016 |
. 2
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49 | 12, 48 | eqssd 3027 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 |
This theorem is referenced by: frecuzrdgdom 9714 |
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