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Theorem fifo 7892
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 4153 . . . . . 6  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  -> 
y  =/=  (/) )
2 intex 4603 . . . . . 6  |-  ( y  =/=  (/)  <->  |^| y  e.  _V )
31, 2sylib 196 . . . . 5  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  ->  |^| y  e.  _V )
43rgen 2824 . . . 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 5707 . . . 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 5801 . . 3  |-  ( F  Fn  ( ( ~P A  i^i  Fin )  \  { (/) } )  <->  F :
( ( ~P A  i^i  Fin )  \  { (/)
} ) -onto-> ran  F
)
97, 8sylib 196 . 2  |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ran 
F )
10 elfi2 7874 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( fi
`  A )  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) x  = 
|^| y ) )
11 vex 3116 . . . . . 6  |-  x  e. 
_V
125elrnmpt 5249 . . . . . 6  |-  ( x  e.  _V  ->  (
x  e.  ran  F  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } ) x  =  |^| y
) )
1311, 12ax-mp 5 . . . . 5  |-  ( x  e.  ran  F  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) x  = 
|^| y )
1410, 13syl6bbr 263 . . . 4  |-  ( A  e.  V  ->  (
x  e.  ( fi
`  A )  <->  x  e.  ran  F ) )
1514eqrdv 2464 . . 3  |-  ( A  e.  V  ->  ( fi `  A )  =  ran  F )
16 foeq3 5793 . . 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 232 1  |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto-> ( fi `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   {csn 4027   |^|cint 4282    |-> cmpt 4505   ran crn 5000    Fn wfn 5583   -onto->wfo 5586   ` cfv 5588   Fincfn 7516   ficfi 7870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-fo 5594  df-fv 5596  df-fi 7871
This theorem is referenced by:  inffien  8444  fictb  8625
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