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	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))

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.

Step	Hyp	Ref	Expression
1		pweq 4318	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
2	1	difeq1d 3889	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝒫 𝑦 ∖ {∅}) = (𝒫 𝑋 ∖ {∅}))
3	2	oveq1d 6857	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴) = ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴))
4	3	raleqdv 3292	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
5	4	anbi2d 622	. . 3 ⊢ (𝑦 = 𝑋 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))))
6		df-acn 9019	. . 3 ⊢ AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))}
7	5, 6	elab2g 3508	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))))
8		elex 3365	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
9		biid 252	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
10	9	baib 531	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
11	8, 10	syl 17	. 2 ⊢ (𝐴 ∈ 𝑊 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
12	7, 11	sylan9bb 505	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))

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