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Theorem dfac8c 7660
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 2283 . 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 7659 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 1528    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   (/)c0 3455   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    We wwe 4351   ` cfv 5255   iota_crio 6297
This theorem is referenced by:  ween  7662  ac5num  7663  dfac8  7761  vitali  18968
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-rmo 2551  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-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  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-fv 5263  df-riota 6304
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