Theorem isacn 9731

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 4546	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
2	1	difeq1d 4052	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝒫 𝑦 ∖ {∅}) = (𝒫 𝑋 ∖ {∅}))
3	2	oveq1d 7270	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴) = ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴))
4	3	raleqdv 3339	. . . 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 628	. . 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 9631	. . 3 ⊢ AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑦 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))}
7	5, 6	elab2g 3604	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))))
8		elex 3440	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
9		biid 260	. . . 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 535	. . 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 509	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 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  AC wacn 9627

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-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  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-iota 6376  df-fv 6426  df-ov 7258  df-acn 9631

This theorem is referenced by:  acni  9732  numacn  9736  finacn  9737  acndom  9738  acndom2  9741  acncc  10127