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Theorem inffien 7877
Description: The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
inffien  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~~  A )

Proof of Theorem inffien
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 infpwfien 7876 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  ~~  A )
2 relen 7050 . . . . . . . . 9  |-  Rel  ~~
32brrelexi 4858 . . . . . . . 8  |-  ( ( ~P A  i^i  Fin )  ~~  A  ->  ( ~P A  i^i  Fin )  e.  _V )
41, 3syl 16 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  e.  _V )
5 difss 3417 . . . . . . 7  |-  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
C_  ( ~P A  i^i  Fin )
6 ssdomg 7089 . . . . . . 7  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  ( ( ( ~P A  i^i  Fin )  \  { (/)
} )  C_  ( ~P A  i^i  Fin )  ->  ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  ( ~P A  i^i  Fin )
) )
74, 5, 6ee10 1382 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  ( ~P A  i^i  Fin )
)
8 domentr 7102 . . . . . 6  |-  ( ( ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  ( ~P A  i^i  Fin )  /\  ( ~P A  i^i  Fin )  ~~  A )  ->  ( ( ~P A  i^i  Fin )  \  { (/) } )  ~<_  A )
97, 1, 8syl2anc 643 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )
10 numdom 7852 . . . . 5  |-  ( ( A  e.  dom  card  /\  ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )  ->  ( ( ~P A  i^i  Fin )  \  { (/) } )  e. 
dom  card )
119, 10syldan 457 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  e.  dom  card )
12 eqid 2387 . . . . . 6  |-  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| x )  =  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x )
1312fifo 7372 . . . . 5  |-  ( A  e.  dom  card  ->  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x ) : ( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ( fi
`  A ) )
1413adantr 452 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| x ) : ( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ( fi
`  A ) )
15 fodomnum 7871 . . . 4  |-  ( ( ( ~P A  i^i  Fin )  \  { (/) } )  e.  dom  card  -> 
( ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} )  |->  |^| x
) : ( ( ~P A  i^i  Fin )  \  { (/) } )
-onto-> ( fi `  A
)  ->  ( fi `  A )  ~<_  ( ( ~P A  i^i  Fin )  \  { (/) } ) ) )
1611, 14, 15sylc 58 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~<_  ( ( ~P A  i^i  Fin )  \  { (/) } ) )
17 domtr 7096 . . 3  |-  ( ( ( fi `  A
)  ~<_  ( ( ~P A  i^i  Fin )  \  { (/) } )  /\  ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )  ->  ( fi `  A )  ~<_  A )
1816, 9, 17syl2anc 643 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~<_  A )
19 fvex 5682 . . 3  |-  ( fi
`  A )  e. 
_V
20 ssfii 7359 . . . 4  |-  ( A  e.  dom  card  ->  A 
C_  ( fi `  A ) )
2120adantr 452 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  C_  ( fi `  A ) )
22 ssdomg 7089 . . 3  |-  ( ( fi `  A )  e.  _V  ->  ( A  C_  ( fi `  A )  ->  A  ~<_  ( fi `  A ) ) )
2319, 21, 22mpsyl 61 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( fi `  A ) )
24 sbth 7163 . 2  |-  ( ( ( fi `  A
)  ~<_  A  /\  A  ~<_  ( fi `  A ) )  ->  ( fi `  A )  ~~  A
)
2518, 23, 24syl2anc 643 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2899    \ cdif 3260    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   |^|cint 3992   class class class wbr 4153    e. cmpt 4207   omcom 4785   dom cdm 4818   -onto->wfo 5392   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043   Fincfn 7045   ficfi 7350   cardccrd 7755
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-seqom 6641  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-oi 7412  df-card 7759  df-acn 7762
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