Theorem dfac8c 9720

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 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 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.

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (℩𝑦 ∈ 𝑥 ∀𝑤 ∈ 𝑥 ¬ 𝑤𝑟𝑦)) = (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (℩𝑦 ∈ 𝑥 ∀𝑤 ∈ 𝑥 ¬ 𝑤𝑟𝑦))
2	1	dfac8clem 9719	1 ⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ∖ cdif 3880  ∅c0 4253  {csn 4558  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153   We wwe 5534  ‘cfv 6418  ℩crio 7211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212

This theorem is referenced by:  ween  9722  ac5num  9723  dfac8  9822  vitali  24682