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Mirrors > Home > ILE Home > Th. List > fihashf1rn | Unicode version |
Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
fihashf1rn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 5212 |
. . 3
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2 | simpl 107 |
. . 3
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3 | fnfi 6636 |
. . 3
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4 | 1, 2, 3 | syl2an2 561 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | f1o2ndf1 5985 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | df-2nd 5904 |
. . . . . . . . 9
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7 | 6 | funmpt2 5047 |
. . . . . . . 8
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8 | f1f 5210 |
. . . . . . . . . . 11
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9 | 8 | anim2i 334 |
. . . . . . . . . 10
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10 | 9 | ancomd 263 |
. . . . . . . . 9
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11 | fex 5516 |
. . . . . . . . 9
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12 | 10, 11 | syl 14 |
. . . . . . . 8
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13 | resfunexg 5510 |
. . . . . . . 8
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14 | 7, 12, 13 | sylancr 405 |
. . . . . . 7
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15 | f1oeq1 5238 |
. . . . . . . . . 10
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16 | 15 | biimpd 142 |
. . . . . . . . 9
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17 | 16 | eqcoms 2091 |
. . . . . . . 8
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18 | 17 | adantl 271 |
. . . . . . 7
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19 | 14, 18 | spcimedv 2705 |
. . . . . 6
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20 | 19 | ex 113 |
. . . . 5
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21 | 20 | com13 79 |
. . . 4
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22 | 5, 21 | mpcom 36 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | impcom 123 |
. 2
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24 | fihasheqf1oi 10184 |
. . . 4
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25 | 24 | ex 113 |
. . 3
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26 | 25 | exlimdv 1747 |
. 2
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27 | 4, 23, 26 | sylc 61 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-iinf 4401 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-addcom 7435 ax-addass 7437 ax-distr 7439 ax-i2m1 7440 ax-0lt1 7441 ax-0id 7443 ax-rnegex 7444 ax-cnre 7446 ax-pre-ltirr 7447 ax-pre-ltwlin 7448 ax-pre-lttrn 7449 ax-pre-ltadd 7451 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3392 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-tr 3935 df-id 4118 df-iord 4191 df-on 4193 df-ilim 4194 df-suc 4196 df-iom 4404 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-2nd 5904 df-recs 6062 df-frec 6148 df-1o 6173 df-er 6282 df-en 6448 df-dom 6449 df-fin 6450 df-pnf 7514 df-mnf 7515 df-xr 7516 df-ltxr 7517 df-le 7518 df-sub 7645 df-neg 7646 df-inn 8413 df-n0 8664 df-z 8741 df-uz 9010 df-ihash 10172 |
This theorem is referenced by: (None) |
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