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Theorem fseqen 8397
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 7515 . 2  |-  ( ( A  X.  A ) 
~~  A  <->  E. f 
f : ( A  X.  A ) -1-1-onto-> A )
2 n0 3787 . 2  |-  ( A  =/=  (/)  <->  E. b  b  e.  A )
3 eeanv 1950 . . 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 8049 . . . . . . 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 5814 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -1-1-onto-> A  ->  f :
( A  X.  A
) -onto-> A )
7 forn 5789 . . . . . . . . 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 3109 . . . . . . . . 9  |-  f  e. 
_V
109rnex 6708 . . . . . . . 8  |-  ran  f  e.  _V
118, 10syl6eqelr 2557 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  A  e.  _V )
12 xpexg 6702 . . . . . . 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 2460 . . . . . . 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 2460 . . . . . . 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 8395 . . . . . 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 7525 . . . . . 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 8396 . . . . . 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 7627 . . . . 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 1694 . . 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 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   _Vcvv 3106   (/)c0 3778   {csn 4020   <.cop 4026   U_ciun 4318   class class class wbr 4440    |-> cmpt 4498   suc csuc 4873    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994   -1-1->wf1 5576   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   omcom 6671  seq𝜔cseqom 7102    ^m cmap 7410    ~~ cen 7503    ~<_ cdom 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-seqom 7103  df-1o 7120  df-map 7412  df-en 7507  df-dom 7508
This theorem is referenced by:  infpwfien  8432
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