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Mirrors > Home > ILE Home > Th. List > hashcl | Unicode version |
Description: Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
Ref | Expression |
---|---|
hashcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6534 |
. . 3
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2 | 1 | biimpi 119 |
. 2
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3 | simprl 499 |
. . . 4
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4 | simprr 500 |
. . . . 5
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5 | 4 | ensymd 6556 |
. . . 4
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6 | hashennn 10251 |
. . . 4
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7 | 3, 5, 6 | syl2anc 404 |
. . 3
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8 | 0zd 8825 |
. . . . . 6
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9 | eqid 2089 |
. . . . . 6
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10 | id 19 |
. . . . . 6
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11 | 8, 9, 10 | frec2uzuzd 9872 |
. . . . 5
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12 | nn0uz 9116 |
. . . . 5
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13 | 11, 12 | syl6eleqr 2182 |
. . . 4
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14 | 3, 13 | syl 14 |
. . 3
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15 | 7, 14 | eqeltrd 2165 |
. 2
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16 | 2, 15 | rexlimddv 2496 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-recs 6086 df-frec 6172 df-er 6308 df-en 6514 df-dom 6515 df-fin 6516 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-n0 8737 df-z 8814 df-uz 9083 df-ihash 10247 |
This theorem is referenced by: hashfiv01gt1 10253 filtinf 10263 isfinite4im 10264 fihashneq0 10266 hashnncl 10267 fihashssdif 10289 hashdifpr 10291 hashxp 10297 zfz1isolemsplit 10306 zfz1isolemiso 10307 zfz1isolem1 10308 fz1f1o 10827 fsumconst 10911 hashiun 10935 hash2iun1dif1 10937 phival 11530 phicl2 11531 phiprmpw 11539 |
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