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Theorem acni 7688
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 4210 . . . . 5  |-  ( X  e. AC  A  ->  ~P X  e.  _V )
2 difexg 4178 . . . . 5  |-  ( ~P X  e.  _V  ->  ( ~P X  \  { (/)
} )  e.  _V )
31, 2syl 15 . . . 4  |-  ( X  e. AC  A  ->  ( ~P X  \  { (/) } )  e.  _V )
4 acnrcl 7685 . . . 4  |-  ( X  e. AC  A  ->  A  e. 
_V )
5 elmapg 6801 . . . 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 7687 . . . . 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 5540 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1312eleq2d 2363 . . . . 5  |-  ( f  =  F  ->  (
( g `  x
)  e.  ( f `
 x )  <->  ( g `  x )  e.  ( F `  x ) ) )
1413ralbidv 2576 . . . 4  |-  ( f  =  F  ->  ( A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1514exbidv 1616 . . 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 2893 . 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 1531    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   (/)c0 3468   ~Pcpw 3638   {csn 3653   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788  AC wacn 7587
This theorem is referenced by:  acni2  7689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-acn 7591
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