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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 6897 | . . . 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 2728 | . . . . . 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 14326 | . . . . 5 β’ (π΄ β Fin β (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β―βπ΄)) = (cardβπ΄)) |
4 | 2 | hashginv 14326 | . . . . 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 2742 | . . . 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 6897 | . . . 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 14325 | . . . . 5 β’ (π΄ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = (β―βπ΄)) |
9 | 2 | hashgval 14325 | . . . . 5 β’ (π΅ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) = (β―βπ΅)) |
10 | 8, 9 | eqeqan12d 2742 | . . . 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 9972 | . . 3 β’ (π΄ β Fin β π΄ β dom card) | |
14 | finnum 9972 | . . 3 β’ (π΅ β Fin β π΅ β dom card) | |
15 | carden2 10011 | . . 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 1534 β wcel 2099 Vcvv 3471 class class class wbr 5148 β¦ cmpt 5231 β‘ccnv 5677 dom cdm 5678 βΎ cres 5680 βcfv 6548 (class class class)co 7420 Οcom 7870 reccrdg 8430 β cen 8961 Fincfn 8964 cardccrd 9959 0cc0 11139 1c1 11140 + caddc 11142 β―chash 14322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-hash 14323 |
This theorem is referenced by: hasheni 14340 hasheqf1o 14341 isfinite4 14354 hasheq0 14355 hashsng 14361 hashen1 14362 hashsdom 14373 hash1snb 14411 hashxplem 14425 hashmap 14427 hashpw 14428 hashbclem 14444 phphashd 14460 hash2pr 14463 pr2pwpr 14473 hash3tr 14484 isercolllem2 15645 isercoll 15647 summolem3 15693 mertenslem1 15863 prodmolem3 15910 bpolylem 16025 hashdvds 16744 crth 16747 phimullem 16748 eulerth 16752 4sqlem11 16924 lagsubg2 19149 dfod2 19519 sylow1lem2 19554 sylow2alem2 19573 slwhash 19579 sylow2 19581 sylow3lem1 19582 cyggenod 19839 lt6abl 19850 ablfac1c 20028 ablfac1eu 20030 ablfaclem3 20044 fta1blem 26118 vieta1 26260 isppw 27059 clwlknon2num 30191 numclwlk1lem2 30193 fisshasheq 34724 derangen2 34784 erdsze2lem1 34813 erdsze2lem2 34814 poimirlem9 37102 poimirlem25 37118 poimirlem26 37119 poimirlem27 37120 poimirlem28 37121 eldioph2lem1 42180 frlmpwfi 42522 isnumbasgrplem3 42529 idomsubgmo 42621 |
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