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Theorem isacn 8905
 Description: The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
isacn ((𝑋𝑉𝐴𝑊) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
Distinct variable groups:   𝑓,𝑔,𝑥,𝐴   𝑓,𝑋,𝑔,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑓,𝑔)   𝑊(𝑥,𝑓,𝑔)

Proof of Theorem isacn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pweq 4194 . . . . . . 7 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
21difeq1d 3760 . . . . . 6 (𝑦 = 𝑋 → (𝒫 𝑦 ∖ {∅}) = (𝒫 𝑋 ∖ {∅}))
32oveq1d 6705 . . . . 5 (𝑦 = 𝑋 → ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴) = ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴))
43raleqdv 3174 . . . 4 (𝑦 = 𝑋 → (∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
54anbi2d 740 . . 3 (𝑦 = 𝑋 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))))
6 df-acn 8806 . . 3 AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))}
75, 6elab2g 3385 . 2 (𝑋𝑉 → (𝑋AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))))
8 elex 3243 . . 3 (𝐴𝑊𝐴 ∈ V)
9 biid 251 . . . 4 ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
109baib 964 . . 3 (𝐴 ∈ V → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
118, 10syl 17 . 2 (𝐴𝑊 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
127, 11sylan9bb 736 1 ((𝑋𝑉𝐴𝑊) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∖ cdif 3604  ∅c0 3948  𝒫 cpw 4191  {csn 4210  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  AC wacn 8802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-acn 8806 This theorem is referenced by:  acni  8906  numacn  8910  finacn  8911  acndom  8912  acndom2  8915  acncc  9300
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