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Theorem fseqen 7834
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 7046 . 2  |-  ( ( A  X.  A ) 
~~  A  <->  E. f 
f : ( A  X.  A ) -1-1-onto-> A )
2 n0 3573 . 2  |-  ( A  =/=  (/)  <->  E. b  b  e.  A )
3 eeanv 1926 . . 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 7524 . . . . . . 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 5614 . . . . . . . . 9  |-  ( f : ( A  X.  A ) -1-1-onto-> A  ->  f :
( A  X.  A
) -onto-> A )
7 forn 5589 . . . . . . . . 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 2895 . . . . . . . . 9  |-  f  e. 
_V
109rnex 5066 . . . . . . . 8  |-  ran  f  e.  _V
118, 10syl6eqelr 2469 . . . . . . 7  |-  ( ( f : ( A  X.  A ) -1-1-onto-> A  /\  b  e.  A )  ->  A  e.  _V )
12 xpexg 4922 . . . . . . 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 2380 . . . . . . 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 2380 . . . . . . 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 7832 . . . . . 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 7056 . . . . . 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 7833 . . . . . 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 7156 . . . . 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 1642 . . 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 1547    = wceq 1649    e. wcel 1717    =/= wne 2543   _Vcvv 2892   (/)c0 3564   {csn 3750   <.cop 3753   U_ciun 4028   class class class wbr 4146    e. cmpt 4200   suc csuc 4517   omcom 4778    X. cxp 4809   dom cdm 4811   ran crn 4812    |` cres 4813   -1-1->wf1 5384   -onto->wfo 5385   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015  seq𝜔cseqom 6633    ^m cmap 6947    ~~ cen 7035    ~<_ cdom 7036
This theorem is referenced by:  infpwfien  7869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-seqom 6634  df-1o 6653  df-map 6949  df-en 7039  df-dom 7040
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