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Theorem inffien 7706
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 7705 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  ~~  A )
2 relen 6884 . . . . . . . . 9  |-  Rel  ~~
32brrelexi 4745 . . . . . . . 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 3316 . . . . . . 7  |-  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
C_  ( ~P A  i^i  Fin )
6 ssdomg 6923 . . . . . . 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 6936 . . . . . 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 7681 . . . . 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 2296 . . . . . 6  |-  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| x )  =  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x )
1312fifo 7201 . . . . 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 7700 . . . 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 6930 . . 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 5555 . . 3  |-  ( fi
`  A )  e. 
_V
20 ssfii 7188 . . . 4  |-  ( A  e.  dom  card  ->  A 
C_  ( fi `  A ) )
2120adantr 451 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  C_  ( fi `  A ) )
22 ssdomg 6923 . . 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 6997 . 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 1696   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   |^|cint 3878   class class class wbr 4039    e. cmpt 4093   omcom 4672   dom cdm 4705   -onto->wfo 5269   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879   ficfi 7180   cardccrd 7584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-oi 7241  df-card 7588  df-acn 7591
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