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Mirrors > Home > MPE Home > Th. List > hashen | Structured version Visualization version GIF version |
Description: Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashen | β’ ((π΄ β Fin β§ π΅ β Fin) β ((β―βπ΄) = (β―βπ΅) β π΄ β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6884 | . . . 4 β’ ((β―βπ΄) = (β―βπ΅) β (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΄)) = (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΅))) | |
2 | eqid 2726 | . . . . . 6 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) | |
3 | 2 | hashginv 14297 | . . . . 5 β’ (π΄ β Fin β (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΄)) = (cardβπ΄)) |
4 | 2 | hashginv 14297 | . . . . 5 β’ (π΅ β Fin β (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΅)) = (cardβπ΅)) |
5 | 3, 4 | eqeqan12d 2740 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β ((β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΄)) = (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΅)) β (cardβπ΄) = (cardβπ΅))) |
6 | 1, 5 | imbitrid 243 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin) β ((β―βπ΄) = (β―βπ΅) β (cardβπ΄) = (cardβπ΅))) |
7 | fveq2 6884 | . . . 4 β’ ((cardβπ΄) = (cardβπ΅) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅))) | |
8 | 2 | hashgval 14296 | . . . . 5 β’ (π΄ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = (β―βπ΄)) |
9 | 2 | hashgval 14296 | . . . . 5 β’ (π΅ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) = (β―βπ΅)) |
10 | 8, 9 | eqeqan12d 2740 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β (((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) β (β―βπ΄) = (β―βπ΅))) |
11 | 7, 10 | imbitrid 243 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin) β ((cardβπ΄) = (cardβπ΅) β (β―βπ΄) = (β―βπ΅))) |
12 | 6, 11 | impbid 211 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin) β ((β―βπ΄) = (β―βπ΅) β (cardβπ΄) = (cardβπ΅))) |
13 | finnum 9942 | . . 3 β’ (π΄ β Fin β π΄ β dom card) | |
14 | finnum 9942 | . . 3 β’ (π΅ β Fin β π΅ β dom card) | |
15 | carden2 9981 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) | |
16 | 13, 14, 15 | syl2an 595 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
17 | 12, 16 | bitrd 279 | 1 β’ ((π΄ β Fin β§ π΅ β Fin) β ((β―βπ΄) = (β―βπ΅) β π΄ β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 class class class wbr 5141 β¦ cmpt 5224 β‘ccnv 5668 dom cdm 5669 βΎ cres 5671 βcfv 6536 (class class class)co 7404 Οcom 7851 reccrdg 8407 β cen 8935 Fincfn 8938 cardccrd 9929 0cc0 11109 1c1 11110 + caddc 11112 β―chash 14293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-hash 14294 |
This theorem is referenced by: hasheni 14311 hasheqf1o 14312 isfinite4 14325 hasheq0 14326 hashsng 14332 hashen1 14333 hashsdom 14344 hash1snb 14382 hashxplem 14396 hashmap 14398 hashpw 14399 hashbclem 14415 phphashd 14431 hash2pr 14434 pr2pwpr 14444 hash3tr 14455 isercolllem2 15616 isercoll 15618 summolem3 15664 mertenslem1 15834 prodmolem3 15881 bpolylem 15996 hashdvds 16715 crth 16718 phimullem 16719 eulerth 16723 4sqlem11 16895 lagsubg2 19118 dfod2 19482 sylow1lem2 19517 sylow2alem2 19536 slwhash 19542 sylow2 19544 sylow3lem1 19545 cyggenod 19802 lt6abl 19813 ablfac1c 19991 ablfac1eu 19993 ablfaclem3 20007 fta1blem 26056 vieta1 26198 isppw 26997 clwlknon2num 30126 numclwlk1lem2 30128 fisshasheq 34633 derangen2 34693 erdsze2lem1 34722 erdsze2lem2 34723 poimirlem9 37008 poimirlem25 37024 poimirlem26 37025 poimirlem27 37026 poimirlem28 37027 eldioph2lem1 42057 frlmpwfi 42399 isnumbasgrplem3 42406 idomsubgmo 42498 |
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