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Theorem ac5 8104
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 8102. (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 5334 . . . 4  |-  ( y  =  A  ->  (
f  Fn  y  <->  f  Fn  A ) )
3 raleq 2736 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y 
( x  =/=  (/)  ->  (
f `  x )  e.  x )  <->  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
42, 3anbi12d 691 . . 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 1612 . 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 7749 . . 3  |-  (CHOICE  <->  A. y E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
76axaci 8095 . 2  |-  E. f
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
81, 5, 7vtocl 2838 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 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   (/)c0 3455    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  ac5g  25075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ac 7743
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