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

Theorem acni 8417
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 4621 . . . . 5  |-  ( X  e. AC  A  ->  ~P X  e.  _V )
2 difexg 4585 . . . . 5  |-  ( ~P X  e.  _V  ->  ( ~P X  \  { (/)
} )  e.  _V )
31, 2syl 16 . . . 4  |-  ( X  e. AC  A  ->  ( ~P X  \  { (/) } )  e.  _V )
4 acnrcl 8414 . . . 4  |-  ( X  e. AC  A  ->  A  e. 
_V )
53, 4elmapd 7426 . . 3  |-  ( X  e. AC  A  ->  ( F  e.  ( ( ~P X  \  { (/) } )  ^m  A )  <-> 
F : A --> ( ~P X  \  { (/) } ) ) )
65biimpar 483 . 2  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  F  e.  ( ( ~P X  \  { (/) } )  ^m  A ) )
7 isacn 8416 . . . . 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 ) ) )
84, 7mpdan 666 . . . 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 ) ) )
98ibi 241 . . 3  |-  ( X  e. AC  A  ->  A. f  e.  ( ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x
)  e.  ( f `
 x ) )
109adantr 463 . 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 ) )
11 fveq1 5847 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1211eleq2d 2524 . . . . 5  |-  ( f  =  F  ->  (
( g `  x
)  e.  ( f `
 x )  <->  ( g `  x )  e.  ( F `  x ) ) )
1312ralbidv 2893 . . . 4  |-  ( f  =  F  ->  ( A. x  e.  A  ( g `  x
)  e.  ( f `
 x )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
1413exbidv 1719 . . 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 ) ) )
1514rspcv 3203 . 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 ) ) )
166, 10, 15sylc 60 1  |-  ( ( X  e. AC  A  /\  F : A --> ( ~P X  \  { (/) } ) )  ->  E. g A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   A.wral 2804   _Vcvv 3106    \ cdif 3458   (/)c0 3783   ~Pcpw 3999   {csn 4016   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412  AC wacn 8310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-acn 8314
This theorem is referenced by:  acni2  8418
  Copyright terms: Public domain W3C validator