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Theorem ac5 8072
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 8070. (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
StepHypRef Expression
1 ac5.1 . 2  |-  A  e. 
_V
2 fneq2 5272 . . . 4  |-  ( y  =  A  ->  (
f  Fn  y  <->  f  Fn  A ) )
3 raleq 2711 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y 
( x  =/=  (/)  ->  (
f `  x )  e.  x )  <->  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
42, 3anbi12d 694 . . 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 2006 . 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 7717 . . 3  |-  (CHOICE  <->  A. y E. f ( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
76axaci 8063 . 2  |-  E. f
( f  Fn  y  /\  A. x  e.  y  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
81, 5, 7vtocl 2813 1  |-  E. f
( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   _Vcvv 2763   (/)c0 3430    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  ac5g  24442
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-ac2 8057
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ac 7711
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