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Theorem dfac8c 7916
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 2438 . 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 7915 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 1551    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319   (/)c0 3630   {csn 3816   U.cuni 4017   class class class wbr 4214    e. cmpt 4268    We wwe 4542   ` cfv 5456   iota_crio 6544
This theorem is referenced by:  ween  7918  ac5num  7919  dfac8  8017  vitali  19507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-riota 6551
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