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Theorem dfac8c 9969
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 (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 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 2736 . 2 (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (𝑦𝑥𝑤𝑥 ¬ 𝑤𝑟𝑦)) = (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (𝑦𝑥𝑤𝑥 ¬ 𝑤𝑟𝑦))
21dfac8clem 9968 1 (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wex 1781  wcel 2106  wne 2943  wral 3064  cdif 3907  c0 4282  {csn 4586   cuni 4865   class class class wbr 5105  cmpt 5188   We wwe 5587  cfv 6496  crio 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313
This theorem is referenced by:  ween  9971  ac5num  9972  dfac8  10071  vitali  24977
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