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Mirrors > Home > ILE Home > Th. List > rexuz3 | Unicode version |
Description: Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.) |
Ref | Expression |
---|---|
rexuz3.1 |
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Ref | Expression |
---|---|
rexuz3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . 5
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2 | 1 | rgen 2540 |
. . . 4
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3 | fveq2 5527 |
. . . . . . 7
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4 | rexuz3.1 |
. . . . . . 7
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5 | 3, 4 | eqtr4di 2238 |
. . . . . 6
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6 | 5 | raleqdv 2689 |
. . . . 5
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7 | 6 | rspcev 2853 |
. . . 4
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8 | 2, 7 | mpan2 425 |
. . 3
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9 | 8 | biantrurd 305 |
. 2
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10 | 4 | uztrn2 9558 |
. . . . . . . . . 10
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11 | 10 | a1d 22 |
. . . . . . . . 9
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12 | 11 | ancrd 326 |
. . . . . . . 8
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13 | 12 | ralimdva 2554 |
. . . . . . 7
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14 | eluzelz 9550 |
. . . . . . . 8
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15 | 14, 4 | eleq2s 2282 |
. . . . . . 7
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16 | 13, 15 | jctild 316 |
. . . . . 6
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17 | 16 | imp 124 |
. . . . 5
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18 | uzid 9555 |
. . . . . . 7
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19 | simpl 109 |
. . . . . . . 8
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20 | 19 | ralimi 2550 |
. . . . . . 7
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21 | eleq1 2250 |
. . . . . . . 8
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22 | 21 | rspcva 2851 |
. . . . . . 7
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23 | 18, 20, 22 | syl2an 289 |
. . . . . 6
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24 | simpr 110 |
. . . . . . . 8
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25 | 24 | ralimi 2550 |
. . . . . . 7
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26 | 25 | adantl 277 |
. . . . . 6
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27 | 23, 26 | jca 306 |
. . . . 5
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28 | 17, 27 | impbii 126 |
. . . 4
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29 | 28 | rexbii2 2498 |
. . 3
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30 | rexanuz 11010 |
. . 3
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31 | 29, 30 | bitr2i 185 |
. 2
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32 | 9, 31 | bitr2di 197 |
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-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 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-apti 7939 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 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-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 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: rexanuz2 11013 cau4 11138 clim2 11304 lmbr2 13985 lmff 14020 |
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