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Theorem fifo 7374
Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
Hypothesis
Ref Expression
fifo.1  |-  F  =  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| y )
Assertion
Ref Expression
fifo  |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ( fi `  A ) )
Distinct variable groups:    y, A    y, V
Allowed substitution hint:    F( y)

Proof of Theorem fifo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifsni 3873 . . . . . 6  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  -> 
y  =/=  (/) )
2 intex 4299 . . . . . 6  |-  ( y  =/=  (/)  <->  |^| y  e.  _V )
31, 2sylib 189 . . . . 5  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  ->  |^| y  e.  _V )
43rgen 2716 . . . 4  |-  A. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) |^| y  e.  _V
5 fifo.1 . . . . 5  |-  F  =  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| y )
65fnmpt 5513 . . . 4  |-  ( A. y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) |^| y  e.  _V  ->  F  Fn  ( ( ~P A  i^i  Fin )  \  { (/) } ) )
74, 6mp1i 12 . . 3  |-  ( A  e.  V  ->  F  Fn  ( ( ~P A  i^i  Fin )  \  { (/)
} ) )
8 dffn4 5601 . . 3  |-  ( F  Fn  ( ( ~P A  i^i  Fin )  \  { (/) } )  <->  F :
( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ran  F
)
97, 8sylib 189 . 2  |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ran 
F )
10 elfi2 7356 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( fi
`  A )  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) x  = 
|^| y ) )
11 vex 2904 . . . . . 6  |-  x  e. 
_V
125elrnmpt 5059 . . . . . 6  |-  ( x  e.  _V  ->  (
x  e.  ran  F  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) x  =  |^| y
) )
1311, 12ax-mp 8 . . . . 5  |-  ( x  e.  ran  F  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) x  = 
|^| y )
1410, 13syl6bbr 255 . . . 4  |-  ( A  e.  V  ->  (
x  e.  ( fi
`  A )  <->  x  e.  ran  F ) )
1514eqrdv 2387 . . 3  |-  ( A  e.  V  ->  ( fi `  A )  =  ran  F )
16 foeq3 5593 . . 3  |-  ( ( fi `  A )  =  ran  F  -> 
( F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ( fi `  A )  <->  F :
( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ran  F
) )
1715, 16syl 16 . 2  |-  ( A  e.  V  ->  ( F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ( fi `  A )  <-> 
F : ( ( ~P A  i^i  Fin )  \  { (/) } )
-onto->
ran  F ) )
189, 17mpbird 224 1  |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ( fi `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   _Vcvv 2901    \ cdif 3262    i^i cin 3264   (/)c0 3573   ~Pcpw 3744   {csn 3759   |^|cint 3994    e. cmpt 4209   ran crn 4821    Fn wfn 5391   -onto->wfo 5394   ` cfv 5396   Fincfn 7047   ficfi 7352
This theorem is referenced by:  inffien  7879  fictb  8060
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-fo 5402  df-fv 5404  df-fi 7353
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