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Theorem acni 7672
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 4194 . . . . 5  |-  ( X  e. AC  A  ->  ~P X  e.  _V )
2 difexg 4162 . . . . 5  |-  ( ~P X  e.  _V  ->  ( ~P X  \  { (/)
} )  e.  _V )
31, 2syl 15 . . . 4  |-  ( X  e. AC  A  ->  ( ~P X  \  { (/) } )  e.  _V )
4 acnrcl 7669 . . . 4  |-  ( X  e. AC  A  ->  A  e. 
_V )
5 elmapg 6785 . . . 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 7671 . . . . 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 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1312eleq2d 2350 . . . . 5  |-  ( f  =  F  ->  (
( g `  x
)  e.  ( f `
 x )  <->  ( g `  x )  e.  ( F `  x ) ) )
1413ralbidv 2563 . . . 4  |-  ( f  =  F  ->  ( A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1514exbidv 1612 . . 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 2880 . 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 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772  AC wacn 7571
This theorem is referenced by:  acni2  7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-acn 7575
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