MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isacn Structured version   Visualization version   GIF version

Theorem isacn 9118
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 4318 . . . . . . 7 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
21difeq1d 3889 . . . . . 6 (𝑦 = 𝑋 → (𝒫 𝑦 ∖ {∅}) = (𝒫 𝑋 ∖ {∅}))
32oveq1d 6857 . . . . 5 (𝑦 = 𝑋 → ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴) = ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴))
43raleqdv 3292 . . . 4 (𝑦 = 𝑋 → (∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
54anbi2d 622 . . 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 9019 . . 3 AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))}
75, 6elab2g 3508 . 2 (𝑋𝑉 → (𝑋AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))))
8 elex 3365 . . 3 (𝐴𝑊𝐴 ∈ V)
9 biid 252 . . . 4 ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
109baib 531 . . 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 505 1 ((𝑋𝑉𝐴𝑊) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  Vcvv 3350  cdif 3729  c0 4079  𝒫 cpw 4315  {csn 4334  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  AC wacn 9015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-ov 6845  df-acn 9019
This theorem is referenced by:  acni  9119  numacn  9123  finacn  9124  acndom  9125  acndom2  9128  acncc  9515
  Copyright terms: Public domain W3C validator