MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fifo Structured version   Unicode version

Theorem fifo 7429
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 3920 . . . . . 6  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  -> 
y  =/=  (/) )
2 intex 4348 . . . . . 6  |-  ( y  =/=  (/)  <->  |^| y  e.  _V )
31, 2sylib 189 . . . . 5  |-  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  ->  |^| y  e.  _V )
43rgen 2763 . . . 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 5563 . . . 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 5651 . . 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 7411 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( fi
`  A )  <->  E. y  e.  ( ( ~P A  i^i  Fin )  \  { (/)
} ) x  = 
|^| y ) )
11 vex 2951 . . . . . 6  |-  x  e. 
_V
125elrnmpt 5109 . . . . . 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 2433 . . 3  |-  ( A  e.  V  ->  ( fi `  A )  =  ran  F )
16 foeq3 5643 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    i^i cin 3311   (/)c0 3620   ~Pcpw 3791   {csn 3806   |^|cint 4042    e. cmpt 4258   ran crn 4871    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446   Fincfn 7101   ficfi 7407
This theorem is referenced by:  inffien  7934  fictb  8115
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-fo 5452  df-fv 5454  df-fi 7408
  Copyright terms: Public domain W3C validator