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Theorem fseqen 7900
Description: A set that is equinumerous to its cross 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 7109 . 2  |-  ( ( A  X.  A ) 
~~  A  <->  E. f 
f : ( A  X.  A ) -1-1-onto-> A )
2 n0 3629 . 2  |-  ( A  =/=  (/)  <->  E. b  b  e.  A )
3 eeanv 1937 . . 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 7590 . . . . . . 7  |-  om  e.  _V
5 simpl 444 . . . . . . . . 9  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  f : ( A  X.  A ) -1-1-onto-> A )
6 f1ofo 5673 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -1-1-onto-> A  ->  f :
( A  X.  A
) -onto-> A )
7 forn 5648 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -onto-> A  ->  ran  f  =  A
)
85, 6, 73syl 19 . . . . . . . 8  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ran  f  =  A )
9 vex 2951 . . . . . . . . 9  |-  f  e. 
_V
109rnex 5125 . . . . . . . 8  |-  ran  f  e.  _V
118, 10syl6eqelr 2524 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  A  e.  _V )
12 xpexg 4981 . . . . . . 7  |-  ( ( om  e.  _V  /\  A  e.  _V )  ->  ( om  X.  A
)  e.  _V )
134, 11, 12sylancr 645 . . . . . 6  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  ( om  X.  A
)  e.  _V )
14 simpr 448 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  b  e.  A )
15 eqid 2435 . . . . . . 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 2435 . . . . . . 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 7898 . . . . . 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 7119 . . . . . 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 58 . . . . 5  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~<_  ( om  X.  A ) )
20 fseqdom 7899 . . . . . 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 7219 . . . . 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 643 . . . 4  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  U_ n  e.  om  ( A  ^m  n
)  ~~  ( om  X.  A ) )
2423exlimivv 1645 . . 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 205 . 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 466 1  |-  ( ( ( A  X.  A
)  ~~  A  /\  A  =/=  (/) )  ->  U_ n  e.  om  ( A  ^m  n )  ~~  ( om  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   {csn 3806   <.cop 3809   U_ciun 4085   class class class wbr 4204    e. cmpt 4258   suc csuc 4575   omcom 4837    X. cxp 4868   dom cdm 4870   ran crn 4871    |` cres 4872   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075  seq𝜔cseqom 6696    ^m cmap 7010    ~~ cen 7098    ~<_ cdom 7099
This theorem is referenced by:  infpwfien  7935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-map 7012  df-en 7102  df-dom 7103
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