MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-acn Unicode version

Definition df-acn 7572
Description: Define a local and length-limited version of the axiom of choice. The definition of the predicate 
X  e. AC  A is that for all families of nonempty subsets of  X indexed on  A (i.e. functions  A --> ~P X  \  { (/) }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df-acn  |- AC  A  =  { x  |  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  A
) E. g A. y  e.  A  (
g `  y )  e.  ( f `  y
) ) }
Distinct variable group:    f, g, x, y, A

Detailed syntax breakdown of Definition df-acn
StepHypRef Expression
1 cA . . 3  class  A
21wacn 7568 . 2  class AC  A
3 cvv 2791 . . . . 5  class  _V
41, 3wcel 1687 . . . 4  wff  A  e. 
_V
5 vy . . . . . . . . . 10  set  y
65cv 1624 . . . . . . . . 9  class  y
7 vg . . . . . . . . . 10  set  g
87cv 1624 . . . . . . . . 9  class  g
96, 8cfv 5223 . . . . . . . 8  class  ( g `
 y )
10 vf . . . . . . . . . 10  set  f
1110cv 1624 . . . . . . . . 9  class  f
126, 11cfv 5223 . . . . . . . 8  class  ( f `
 y )
139, 12wcel 1687 . . . . . . 7  wff  ( g `
 y )  e.  ( f `  y
)
1413, 5, 1wral 2546 . . . . . 6  wff  A. y  e.  A  ( g `  y )  e.  ( f `  y )
1514, 7wex 1530 . . . . 5  wff  E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y )
16 vx . . . . . . . . 9  set  x
1716cv 1624 . . . . . . . 8  class  x
1817cpw 3628 . . . . . . 7  class  ~P x
19 c0 3458 . . . . . . . 8  class  (/)
2019csn 3643 . . . . . . 7  class  { (/) }
2118, 20cdif 3152 . . . . . 6  class  ( ~P x  \  { (/) } )
22 cmap 6769 . . . . . 6  class  ^m
2321, 1, 22co 5821 . . . . 5  class  ( ( ~P x  \  { (/)
} )  ^m  A
)
2415, 10, 23wral 2546 . . . 4  wff  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y )
254, 24wa 360 . . 3  wff  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y )  e.  ( f `  y ) )
2625, 16cab 2272 . 2  class  { x  |  ( A  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) ) }
272, 26wceq 1625 1  wff AC  A  =  { x  |  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  A
) E. g A. y  e.  A  (
g `  y )  e.  ( f `  y
) ) }
Colors of variables: wff set class
This definition is referenced by:  acnrcl  7666  acneq  7667  isacn  7668
  Copyright terms: Public domain W3C validator