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Theorem acni 7762
Description: The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acni  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  E. g A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) )
Distinct variable groups:    x, g, A    g, F, x    g, X, x

Proof of Theorem acni
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 pwexg 4275 . . . . 5  |-  ( X  e. AC  A  ->  ~P X  e.  _V )
2 difexg 4243 . . . . 5  |-  ( ~P X  e.  _V  ->  ( ~P X  \  { (/)
} )  e.  _V )
31, 2syl 15 . . . 4  |-  ( X  e. AC  A  ->  ( ~P X  \  { (/) } )  e.  _V )
4 acnrcl 7759 . . . 4  |-  ( X  e. AC  A  ->  A  e. 
_V )
5 elmapg 6873 . . . 4  |-  ( ( ( ~P X  \  { (/) } )  e. 
_V  /\  A  e.  _V )  ->  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  <-> 
F : A --> ( ~P X  \  { (/) } ) ) )
63, 4, 5syl2anc 642 . . 3  |-  ( X  e. AC  A  ->  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  <-> 
F : A --> ( ~P X  \  { (/) } ) ) )
76biimpar 471 . 2  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  F  e.  ( ( ~P X  \  { (/) } )  ^m  A ) )
8 isacn 7761 . . . . 5  |-  ( ( X  e. AC  A  /\  A  e.  _V )  ->  ( X  e. AC  A  <->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) ) )
94, 8mpdan 649 . . . 4  |-  ( X  e. AC  A  ->  ( X  e. AC  A  <->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) ) )
109ibi 232 . . 3  |-  ( X  e. AC  A  ->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) )
1110adantr 451 . 2  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) )
12 fveq1 5607 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1312eleq2d 2425 . . . . 5  |-  ( f  =  F  ->  (
( g `  x
)  e.  ( f `
 x )  <->  ( g `  x )  e.  ( F `  x ) ) )
1413ralbidv 2639 . . . 4  |-  ( f  =  F  ->  ( A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1514exbidv 1626 . . 3  |-  ( f  =  F  ->  ( E. g A. x  e.  A  ( g `  x )  e.  ( f `  x )  <->  E. g A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1615rspcv 2956 . 2  |-  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  ->  ( A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  ->  E. g A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
177, 11, 16sylc 56 1  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  E. g A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    \ cdif 3225   (/)c0 3531   ~Pcpw 3701   {csn 3716   -->wf 5333   ` cfv 5337  (class class class)co 5945    ^m cmap 6860  AC wacn 7661
This theorem is referenced by:  acni2  7763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-acn 7665
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