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Theorem inffien 7690
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 7689 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  ~~  A )
2 relen 6868 . . . . . . . . 9  |-  Rel  ~~
32brrelexi 4729 . . . . . . . 8  |-  ( ( ~P A  i^i  Fin )  ~~  A  ->  ( ~P A  i^i  Fin )  e.  _V )
41, 3syl 15 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  e.  _V )
5 difss 3303 . . . . . . 7  |-  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
C_  ( ~P A  i^i  Fin )
6 ssdomg 6907 . . . . . . 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 1366 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  ( ~P A  i^i  Fin )
)
8 domentr 6920 . . . . . 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 642 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )
10 numdom 7665 . . . . 5  |-  ( ( A  e.  dom  card  /\  ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )  ->  ( ( ~P A  i^i  Fin )  \  { (/) } )  e. 
dom  card )
119, 10syldan 456 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  e.  dom  card )
12 eqid 2283 . . . . . 6  |-  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| x )  =  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x )
1312fifo 7185 . . . . 5  |-  ( A  e.  dom  card  ->  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x ) : ( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ( fi
`  A ) )
1413adantr 451 . . . 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 7684 . . . 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 56 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~<_  ( ( ~P A  i^i  Fin )  \  { (/) } ) )
17 domtr 6914 . . 3  |-  ( ( ( fi `  A
)  ~<_  ( ( ~P A  i^i  Fin )  \  { (/) } )  /\  ( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  A )  ->  ( fi `  A )  ~<_  A )
1816, 9, 17syl2anc 642 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~<_  A )
19 fvex 5539 . . 3  |-  ( fi
`  A )  e. 
_V
20 ssfii 7172 . . . 4  |-  ( A  e.  dom  card  ->  A 
C_  ( fi `  A ) )
2120adantr 451 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  C_  ( fi `  A ) )
22 ssdomg 6907 . . 3  |-  ( ( fi `  A )  e.  _V  ->  ( A  C_  ( fi `  A )  ->  A  ~<_  ( fi `  A ) ) )
2319, 21, 22mpsyl 59 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( fi `  A ) )
24 sbth 6981 . 2  |-  ( ( ( fi `  A
)  ~<_  A  /\  A  ~<_  ( fi `  A ) )  ->  ( fi `  A )  ~~  A
)
2518, 23, 24syl2anc 642 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( fi `  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   |^|cint 3862   class class class wbr 4023    e. cmpt 4077   omcom 4656   dom cdm 4689   -onto->wfo 5253   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863   ficfi 7164   cardccrd 7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-oi 7225  df-card 7572  df-acn 7575
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