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Theorem fseqen 8189
Description: A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
fseqen  |-  ( ( ( A  X.  A
)  ~~  A  /\  A  =/=  (/) )  ->  U_ n  e.  om  ( A  ^m  n )  ~~  ( om  X.  A ) )
Distinct variable group:    A, n

Proof of Theorem fseqen
Dummy variables  f 
b  g  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 7311 . 2  |-  ( ( A  X.  A ) 
~~  A  <->  E. f 
f : ( A  X.  A ) -1-1-onto-> A )
2 n0 3641 . 2  |-  ( A  =/=  (/)  <->  E. b  b  e.  A )
3 eeanv 1931 . . 3  |-  ( E. f E. b ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  <->  ( E. f  f : ( A  X.  A
)
-1-1-onto-> A  /\  E. b  b  e.  A ) )
4 omex 7841 . . . . . . 7  |-  om  e.  _V
5 simpl 457 . . . . . . . . 9  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  f : ( A  X.  A ) -1-1-onto-> A )
6 f1ofo 5643 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -1-1-onto-> A  ->  f :
( A  X.  A
) -onto-> A )
7 forn 5618 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -onto-> A  ->  ran  f  =  A
)
85, 6, 73syl 20 . . . . . . . 8  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ran  f  =  A )
9 vex 2970 . . . . . . . . 9  |-  f  e. 
_V
109rnex 6507 . . . . . . . 8  |-  ran  f  e.  _V
118, 10syl6eqelr 2527 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  A  e.  _V )
12 xpexg 6502 . . . . . . 7  |-  ( ( om  e.  _V  /\  A  e.  _V )  ->  ( om  X.  A
)  e.  _V )
134, 11, 12sylancr 663 . . . . . 6  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ( om  X.  A
)  e.  _V )
14 simpr 461 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  b  e.  A )
15 eqid 2438 . . . . . . 7  |- seq𝜔 ( ( k  e. 
_V ,  g  e. 
_V  |->  ( y  e.  ( A  ^m  suc  k )  |->  ( ( g `  ( y  |`  k ) ) f ( y `  k
) ) ) ) ,  { <. (/) ,  b
>. } )  = seq𝜔 ( ( k  e.  _V , 
g  e.  _V  |->  ( y  e.  ( A  ^m  suc  k ) 
|->  ( ( g `  ( y  |`  k
) ) f ( y `  k ) ) ) ) ,  { <. (/) ,  b >. } )
16 eqid 2438 . . . . . . 7  |-  ( x  e.  U_ n  e. 
om  ( A  ^m  n )  |->  <. dom  x ,  ( (seq𝜔 ( ( k  e.  _V , 
g  e.  _V  |->  ( y  e.  ( A  ^m  suc  k ) 
|->  ( ( g `  ( y  |`  k
) ) f ( y `  k ) ) ) ) ,  { <. (/) ,  b >. } ) `  dom  x ) `  x
) >. )  =  ( x  e.  U_ n  e.  om  ( A  ^m  n )  |->  <. dom  x ,  ( (seq𝜔 ( ( k  e.  _V , 
g  e.  _V  |->  ( y  e.  ( A  ^m  suc  k ) 
|->  ( ( g `  ( y  |`  k
) ) f ( y `  k ) ) ) ) ,  { <. (/) ,  b >. } ) `  dom  x ) `  x
) >. )
1711, 14, 5, 15, 16fseqenlem2 8187 . . . . . 6  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ( x  e.  U_ n  e.  om  ( A  ^m  n )  |->  <. dom  x ,  ( (seq𝜔 ( ( k  e.  _V ,  g  e.  _V  |->  ( y  e.  ( A  ^m  suc  k
)  |->  ( ( g `
 ( y  |`  k ) ) f ( y `  k
) ) ) ) ,  { <. (/) ,  b
>. } ) `  dom  x ) `  x
) >. ) : U_ n  e.  om  ( A  ^m  n ) -1-1-> ( om  X.  A ) )
18 f1domg 7321 . . . . . 6  |-  ( ( om  X.  A )  e.  _V  ->  (
( x  e.  U_ n  e.  om  ( A  ^m  n )  |->  <. dom  x ,  ( (seq𝜔 ( ( k  e.  _V ,  g  e.  _V  |->  ( y  e.  ( A  ^m  suc  k
)  |->  ( ( g `
 ( y  |`  k ) ) f ( y `  k
) ) ) ) ,  { <. (/) ,  b
>. } ) `  dom  x ) `  x
) >. ) : U_ n  e.  om  ( A  ^m  n ) -1-1-> ( om  X.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~<_  ( om  X.  A ) ) )
1913, 17, 18sylc 60 . . . . 5  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~<_  ( om  X.  A ) )
20 fseqdom 8188 . . . . . 6  |-  ( A  e.  _V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
2111, 20syl 16 . . . . 5  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ( om  X.  A
)  ~<_  U_ n  e.  om  ( A  ^m  n
) )
22 sbth 7423 . . . . 5  |-  ( (
U_ n  e.  om  ( A  ^m  n
)  ~<_  ( om  X.  A )  /\  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )  ->  U_ n  e.  om  ( A  ^m  n
)  ~~  ( om  X.  A ) )
2319, 21, 22syl2anc 661 . . . 4  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~~  ( om  X.  A ) )
2423exlimivv 1689 . . 3  |-  ( E. f E. b ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~~  ( om  X.  A ) )
253, 24sylbir 213 . 2  |-  ( ( E. f  f : ( A  X.  A
)
-1-1-onto-> A  /\  E. b  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~~  ( om  X.  A ) )
261, 2, 25syl2anb 479 1  |-  ( ( ( A  X.  A
)  ~~  A  /\  A  =/=  (/) )  ->  U_ n  e.  om  ( A  ^m  n )  ~~  ( om  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   _Vcvv 2967   (/)c0 3632   {csn 3872   <.cop 3878   U_ciun 4166   class class class wbr 4287    e. cmpt 4345   suc csuc 4716    X. cxp 4833   dom cdm 4835   ran crn 4836    |` cres 4837   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   omcom 6471  seq𝜔cseqom 6894    ^m cmap 7206    ~~ cen 7299    ~<_ cdom 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-seqom 6895  df-1o 6912  df-map 7208  df-en 7303  df-dom 7304
This theorem is referenced by:  infpwfien  8224
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