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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 5169 |
. . 3
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2 | simpl 107 |
. . 3
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3 | fnfi 6574 |
. . 3
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4 | 1, 2, 3 | syl2an2 559 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | f1o2ndf1 5931 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | df-2nd 5850 |
. . . . . . . . 9
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7 | 6 | funmpt2 5009 |
. . . . . . . 8
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8 | f1f 5167 |
. . . . . . . . . . 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 5467 |
. . . . . . . . 9
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12 | 10, 11 | syl 14 |
. . . . . . . 8
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13 | resfunexg 5461 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 7, 12, 13 | sylancr 405 |
. . . . . . 7
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15 | f1oeq1 5195 |
. . . . . . . . . 10
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16 | 15 | biimpd 142 |
. . . . . . . . 9
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17 | 16 | eqcoms 2088 |
. . . . . . . 8
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18 | 17 | adantl 271 |
. . . . . . 7
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19 | 14, 18 | spcimedv 2697 |
. . . . . 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 10045 |
. . . 4
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25 | 24 | ex 113 |
. . 3
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26 | 25 | exlimdv 1744 |
. 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 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-if 3377 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-2nd 5850 df-recs 6005 df-frec 6091 df-1o 6116 df-er 6225 df-en 6391 df-dom 6392 df-fin 6393 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 df-ihash 10033 |
This theorem is referenced by: (None) |
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