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Theorem dfac8c 7914
Description: If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
dfac8c  |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) ) )
Distinct variable groups:    f, r,
z, A    B, r
Allowed substitution hints:    B( z, f)

Proof of Theorem dfac8c
Dummy variables  w  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . 2  |-  ( x  e.  ( A  \  { (/) } )  |->  (
iota_ y  e.  x A. w  e.  x  -.  w r y ) )  =  ( x  e.  ( A  \  { (/) } )  |->  (
iota_ y  e.  x A. w  e.  x  -.  w r y ) )
21dfac8clem 7913 1  |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1550    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317   (/)c0 3628   {csn 3814   U.cuni 4015   class class class wbr 4212    e. cmpt 4266    We wwe 4540   ` cfv 5454   iota_crio 6542
This theorem is referenced by:  ween  7916  ac5num  7917  dfac8  8015  vitali  19505
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-riota 6549
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