Theorem acni 9967

Description: The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acni	⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝑔,𝐴   𝑔,𝐹,𝑥   𝑔,𝑋,𝑥

Proof of Theorem acni

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6839	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
2	1	eleq2d 2822	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
3	2	ralbidv 3160	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
4	3	exbidv 1923	. 2 ⊢ (𝑓 = 𝐹 → (∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
5		acnrcl 9964	. . . . 5 ⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
6		isacn 9966	. . . . 5 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
7	5, 6	mpdan 688	. . . 4 ⊢ (𝑋 ∈ AC 𝐴 → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
8	7	ibi 267	. . 3 ⊢ (𝑋 ∈ AC 𝐴 → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
9	8	adantr 480	. 2 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
10		pwexg 5320	. . . . 5 ⊢ (𝑋 ∈ AC 𝐴 → 𝒫 𝑋 ∈ V)
11	10	difexd 5272	. . . 4 ⊢ (𝑋 ∈ AC 𝐴 → (𝒫 𝑋 ∖ {∅}) ∈ V)
12	11, 5	elmapd 8787	. . 3 ⊢ (𝑋 ∈ AC 𝐴 → (𝐹 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) ↔ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})))
13	12	biimpar 477	. 2 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → 𝐹 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴))
14	4, 9, 13	rspcdva 3565	1 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  AC wacn 9862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-acn 9866

This theorem is referenced by:  acni2  9968