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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 6757 |
. . 3
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2 | 1 | biimpi 120 |
. 2
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3 | simprl 529 |
. . . 4
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4 | simprr 531 |
. . . . 5
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5 | 4 | ensymd 6779 |
. . . 4
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6 | hashennn 10752 |
. . . 4
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7 | 3, 5, 6 | syl2anc 411 |
. . 3
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8 | 0zd 9260 |
. . . . . 6
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9 | eqid 2177 |
. . . . . 6
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10 | id 19 |
. . . . . 6
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11 | 8, 9, 10 | frec2uzuzd 10396 |
. . . . 5
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12 | nn0uz 9557 |
. . . . 5
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13 | 11, 12 | eleqtrrdi 2271 |
. . . 4
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14 | 3, 13 | syl 14 |
. . 3
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15 | 7, 14 | eqeltrd 2254 |
. 2
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16 | 2, 15 | rexlimddv 2599 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-addcom 7907 ax-addass 7909 ax-distr 7911 ax-i2m1 7912 ax-0lt1 7913 ax-0id 7915 ax-rnegex 7916 ax-cnre 7918 ax-pre-ltirr 7919 ax-pre-ltwlin 7920 ax-pre-lttrn 7921 ax-pre-ltadd 7923 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-recs 6302 df-frec 6388 df-er 6531 df-en 6737 df-dom 6738 df-fin 6739 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-sub 8125 df-neg 8126 df-inn 8915 df-n0 9172 df-z 9249 df-uz 9524 df-ihash 10748 |
This theorem is referenced by: hashfiv01gt1 10754 filtinf 10763 isfinite4im 10764 fihashneq0 10766 hashnncl 10767 fihashssdif 10790 hashdifpr 10792 hashxp 10798 zfz1isolemsplit 10810 zfz1isolemiso 10811 zfz1isolem1 10812 fz1f1o 11375 fsumconst 11454 hashiun 11478 hash2iun1dif1 11480 fprodconst 11620 phival 12204 phicl2 12205 phiprmpw 12213 sumhashdc 12336 hashfinmndnn 12764 |
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