Theorem acni 8868

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		pwexg 4850	. . . . 5 ⊢ (𝑋 ∈ AC 𝐴 → 𝒫 𝑋 ∈ V)
2		difexg 4808	. . . . 5 ⊢ (𝒫 𝑋 ∈ V → (𝒫 𝑋 ∖ {∅}) ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝑋 ∈ AC 𝐴 → (𝒫 𝑋 ∖ {∅}) ∈ V)
4		acnrcl 8865	. . . 4 ⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
5	3, 4	elmapd 7871	. . 3 ⊢ (𝑋 ∈ AC 𝐴 → (𝐹 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})))
6	5	biimpar 502	. 2 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → 𝐹 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴))
7		isacn 8867	. . . . 5 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
8	4, 7	mpdan 702	. . . 4 ⊢ (𝑋 ∈ AC 𝐴 → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
9	8	ibi 256	. . 3 ⊢ (𝑋 ∈ AC 𝐴 → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
10	9	adantr 481	. 2 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
11		fveq1 6190	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12	11	eleq2d 2687	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
13	12	ralbidv 2986	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
14	13	exbidv 1850	. . 3 ⊢ (𝑓 = 𝐹 → (∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
15	14	rspcv 3305	. 2 ⊢ (𝐹 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴) → (∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
16	6, 10, 15	sylc 65	1 ⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  AC wacn 8764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-acn 8768

This theorem is referenced by:  acni2  8869