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Theorem inffien 7936
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 7935 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ~P A  i^i  Fin )  ~~  A )
2 relen 7106 . . . . . . . . 9  |-  Rel  ~~
32brrelexi 4910 . . . . . . . 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 3466 . . . . . . 7  |-  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
C_  ( ~P A  i^i  Fin )
6 ssdomg 7145 . . . . . . 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 1385 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( ~P A  i^i  Fin )  \  { (/)
} )  ~<_  ( ~P A  i^i  Fin )
)
8 domentr 7158 . . . . . 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 7911 . . . . 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 2435 . . . . . 6  |-  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| x )  =  ( x  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) 
|->  |^| x )
1312fifo 7429 . . . . 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 7930 . . . 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 7152 . . 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 5734 . . 3  |-  ( fi
`  A )  e. 
_V
20 ssfii 7416 . . . 4  |-  ( A  e.  dom  card  ->  A 
C_  ( fi `  A ) )
2120adantr 452 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  C_  ( fi `  A ) )
22 ssdomg 7145 . . 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 7219 . 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 1725   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   |^|cint 4042   class class class wbr 4204    e. cmpt 4258   omcom 4837   dom cdm 4870   -onto->wfo 5444   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101   ficfi 7407   cardccrd 7814
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-rmo 2705  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-int 4043  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-se 4534  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-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-oi 7471  df-card 7818  df-acn 7821
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