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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 6891 | . . . 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 2732 | . . . . . 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 14293 | . . . . 5 β’ (π΄ β Fin β (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΄)) = (cardβπ΄)) |
4 | 2 | hashginv 14293 | . . . . 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 2746 | . . . 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 6891 | . . . 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 14292 | . . . . 5 β’ (π΄ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = (β―βπ΄)) |
9 | 2 | hashgval 14292 | . . . . 5 β’ (π΅ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) = (β―βπ΅)) |
10 | 8, 9 | eqeqan12d 2746 | . . . 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 596 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
17 | 12, 16 | bitrd 278 | 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 396 = wceq 1541 β wcel 2106 Vcvv 3474 class class class wbr 5148 β¦ cmpt 5231 β‘ccnv 5675 dom cdm 5676 βΎ cres 5678 βcfv 6543 (class class class)co 7408 Οcom 7854 reccrdg 8408 β cen 8935 Fincfn 8938 cardccrd 9929 0cc0 11109 1c1 11110 + caddc 11112 β―chash 14289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-hash 14290 |
This theorem is referenced by: hasheni 14307 hasheqf1o 14308 isfinite4 14321 hasheq0 14322 hashsng 14328 hashen1 14329 hashsdom 14340 hash1snb 14378 hashxplem 14392 hashmap 14394 hashpw 14395 hashbclem 14410 phphashd 14426 hash2pr 14429 pr2pwpr 14439 hash3tr 14450 isercolllem2 15611 isercoll 15613 summolem3 15659 mertenslem1 15829 prodmolem3 15876 bpolylem 15991 hashdvds 16707 crth 16710 phimullem 16711 eulerth 16715 4sqlem11 16887 lagsubg2 19070 dfod2 19431 sylow1lem2 19466 sylow2alem2 19485 slwhash 19491 sylow2 19493 sylow3lem1 19494 cyggenod 19751 lt6abl 19762 ablfac1c 19940 ablfac1eu 19942 ablfaclem3 19956 fta1blem 25685 vieta1 25824 isppw 26615 clwlknon2num 29618 numclwlk1lem2 29620 fisshasheq 34099 derangen2 34160 erdsze2lem1 34189 erdsze2lem2 34190 poimirlem9 36492 poimirlem25 36508 poimirlem26 36509 poimirlem27 36510 poimirlem28 36511 eldioph2lem1 41488 frlmpwfi 41830 isnumbasgrplem3 41837 idomsubgmo 41930 |
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