MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac5 Unicode version

Theorem ac5 8346
Description: An Axiom of Choice equivalent: there exists a function 
f (called a choice function) with domain 
A that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that  f be a function is not necessary; see ac4 8344. (Contributed by NM, 29-Aug-1999.)
Hypothesis
Ref Expression
ac5.1  |-  A  e. 
_V
Assertion
Ref Expression
ac5  |-  E. f
( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
Distinct variable group:    x, f, A

Proof of Theorem ac5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ac5.1 . 2  |-  A  e. 
_V
2 fneq2 5526 . . . 4  |-  ( y  =  A  ->  (
f  Fn  y  <->  f  Fn  A ) )
3 raleq 2896 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y 
( x  =/=  (/)  ->  (
f `  x )  e.  x )  <->  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
42, 3anbi12d 692 . . 3  |-  ( y  =  A  ->  (
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) ) )
54exbidv 1636 . 2  |-  ( y  =  A  ->  ( E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )  <->  E. f
( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) ) ) )
6 dfac4 7992 . . 3  |-  (CHOICE  <->  A. y E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
76axaci 8337 . 2  |-  E. f
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
81, 5, 7vtocl 2998 1  |-  E. f
( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948   (/)c0 3620    Fn wfn 5440   ` cfv 5445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-ac2 8332
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ac 7986
  Copyright terms: Public domain W3C validator