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Theorem ac5 8106
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 8104. (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 5336 . . . 4  |-  ( y  =  A  ->  (
f  Fn  y  <->  f  Fn  A ) )
3 raleq 2738 . . . 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 1614 . 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 7751 . . 3  |-  (CHOICE  <->  A. y E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
76axaci 8097 . 2  |-  E. f
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
81, 5, 7vtocl 2840 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 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   _Vcvv 2790   (/)c0 3457    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  ac5g  25086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-ac2 8091
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ac 7745
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