ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hashen Unicode version

Theorem hashen 10246
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.)
Assertion
Ref Expression
hashen  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A
)  =  ( `  B
)  <->  A  ~~  B ) )

Proof of Theorem hashen
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6532 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 271 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6532 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad2antlr 474 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrl 503 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  e.  om )
8 simprl 499 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  e.  om )
9 nneneq 6627 . . . . 5  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~~  m  <->  n  =  m ) )
107, 8, 9syl2anc 404 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( n  ~~  m  <->  n  =  m ) )
11 simplrr 504 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  A  ~~  n )
12 enen1 6610 . . . . . 6  |-  ( A 
~~  n  ->  ( A  ~~  B  <->  n  ~~  B ) )
1311, 12syl 14 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~~  B  <->  n 
~~  B ) )
14 simprr 500 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  B  ~~  m )
15 enen2 6611 . . . . . 6  |-  ( B 
~~  m  ->  (
n  ~~  B  <->  n  ~~  m ) )
1614, 15syl 14 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( n  ~~  B  <->  n 
~~  m ) )
1713, 16bitrd 187 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~~  B  <->  n 
~~  m ) )
1811ensymd 6554 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  ~~  A )
19 hashennn 10242 . . . . . . 7  |-  ( ( n  e.  om  /\  n  ~~  A )  -> 
( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
207, 18, 19syl2anc 404 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
2114ensymd 6554 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  ~~  B )
22 hashennn 10242 . . . . . . 7  |-  ( ( m  e.  om  /\  m  ~~  B )  -> 
( `  B )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
238, 21, 22syl2anc 404 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( `  B )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
2420, 23eqeq12d 2103 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  =  ( `  B
)  <->  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n )  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  m
) ) )
25 0zd 8816 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
0  e.  ZZ )
26 eqid 2089 . . . . . . . 8  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )
2725, 26frec2uzf1od 9867 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) : om -1-1-onto-> ( ZZ>=
`  0 ) )
28 f1of1 5265 . . . . . . 7  |-  (frec ( ( x  e.  ZZ  |->  ( x  +  1
) ) ,  0 ) : om -1-1-onto-> ( ZZ>= `  0 )  -> frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) : om -1-1-> (
ZZ>= `  0 ) )
2927, 28syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) : om -1-1-> (
ZZ>= `  0 ) )
30 f1fveq 5565 . . . . . 6  |-  ( (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) : om -1-1-> (
ZZ>= `  0 )  /\  ( n  e.  om  /\  m  e.  om )
)  ->  ( (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  n
)  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1
) ) ,  0 ) `  m )  <-> 
n  =  m ) )
3129, 7, 8, 30syl12anc 1173 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `
 n )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m )  <->  n  =  m ) )
3224, 31bitrd 187 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  =  ( `  B
)  <->  n  =  m
) )
3310, 17, 323bitr4rd 220 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  =  ( `  B
)  <->  A  ~~  B ) )
346, 33rexlimddv 2494 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( ( `  A )  =  ( `  B )  <->  A  ~~  B ) )
353, 34rexlimddv 2494 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A
)  =  ( `  B
)  <->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   E.wrex 2361   class class class wbr 3851    |-> cmpt 3905   omcom 4418   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028  (class class class)co 5666  freccfrec 6169    ~~ cen 6509   Fincfn 6511   0cc0 7404   1c1 7405    + caddc 7407   ZZcz 8804   ZZ>=cuz 9073  ♯chash 10237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-ihash 10238
This theorem is referenced by:  hasheqf1o  10247  isfinite4im  10255  fihasheq0  10256  hashsng  10260  fihashen1  10261  fihashfn  10262  hashun  10267  hashfz  10283  hashxp  10288  mertenslemi1  10983  hashdvds  11529  crth  11532  phimullem  11533
  Copyright terms: Public domain W3C validator