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Theorem ac5 8290
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 8288. (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 5475 . . . 4  |-  ( y  =  A  ->  (
f  Fn  y  <->  f  Fn  A ) )
3 raleq 2847 . . . 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 1633 . 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 7936 . . 3  |-  (CHOICE  <->  A. y E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
76axaci 8281 . 2  |-  E. f
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
81, 5, 7vtocl 2949 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 1547    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899   (/)c0 3571    Fn wfn 5389   ` cfv 5394
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-ac2 8276
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ac 7930
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