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

Theorem acni 7890
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 4351 . . . . 5  |-  ( X  e. AC  A  ->  ~P X  e.  _V )
2 difexg 4319 . . . . 5  |-  ( ~P X  e.  _V  ->  ( ~P X  \  { (/)
} )  e.  _V )
31, 2syl 16 . . . 4  |-  ( X  e. AC  A  ->  ( ~P X  \  { (/) } )  e.  _V )
4 acnrcl 7887 . . . 4  |-  ( X  e. AC  A  ->  A  e. 
_V )
5 elmapg 6998 . . . 4  |-  ( ( ( ~P X  \  { (/) } )  e. 
_V  /\  A  e.  _V )  ->  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  <-> 
F : A --> ( ~P X  \  { (/) } ) ) )
63, 4, 5syl2anc 643 . . 3  |-  ( X  e. AC  A  ->  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  <-> 
F : A --> ( ~P X  \  { (/) } ) ) )
76biimpar 472 . 2  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  F  e.  ( ( ~P X  \  { (/) } )  ^m  A ) )
8 isacn 7889 . . . . 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 650 . . . 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 233 . . 3  |-  ( X  e. AC  A  ->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) )
1110adantr 452 . 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 5694 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1312eleq2d 2479 . . . . 5  |-  ( f  =  F  ->  (
( g `  x
)  e.  ( f `
 x )  <->  ( g `  x )  e.  ( F `  x ) ) )
1413ralbidv 2694 . . . 4  |-  ( f  =  F  ->  ( A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1514exbidv 1633 . . 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 3016 . 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 58 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    \ cdif 3285   (/)c0 3596   ~Pcpw 3767   {csn 3782   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^m cmap 6985  AC wacn 7789
This theorem is referenced by:  acni2  7891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-acn 7793
  Copyright terms: Public domain W3C validator