Theorem dfacacn 10182

Description: A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfacacn	⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Proof of Theorem dfacacn

Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		acacni 10181	. . . 4 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V)
2	1	elvd 3486	. . 3 ⊢ (CHOICE → AC 𝑥 = V)
3	2	alrimiv 1927	. 2 ⊢ (CHOICE → ∀𝑥AC 𝑥 = V)
4		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	difexi 5330	. . . . . 6 ⊢ (𝑦 ∖ {∅}) ∈ V
6		acneq 10083	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
7	6	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
8	5, 7	spcv 3605	. . . . 5 ⊢ (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
9		vuniex 7759	. . . . . . 7 ⊢ ∪ 𝑦 ∈ V
10		id 22	. . . . . . 7 ⊢ (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
11	9, 10	eleqtrrid 2848	. . . . . 6 ⊢ (AC (𝑦 ∖ {∅}) = V → ∪ 𝑦 ∈ AC (𝑦 ∖ {∅}))
12		eldifi 4131	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ∈ 𝑦)
13		elssuni 4937	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ⊆ ∪ 𝑦)
15		eldifsni 4790	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
16	14, 15	jca 511	. . . . . . 7 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅))
17	16	rgen 3063	. . . . . 6 ⊢ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)
18		acni2 10086	. . . . . 6 ⊢ ((∪ 𝑦 ∈ AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
19	11, 17, 18	sylancl 586	. . . . 5 ⊢ (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
20	4	mptex 7243	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V
21		simpr 484	. . . . . . . . 9 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧)
22		eldifsn 4786	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅))
23	22	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
24		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
25		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))
26		fvex 6919	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑧) ∈ V
27	24, 25, 26	fvmpt 7016	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
28	12, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
29	28	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔‘𝑧) ∈ 𝑧))
30	29	pm5.74i 271	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧))
31		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
32	23, 30, 31	3bitr3i 301	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
33	32	ralbii2 3089	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
34	21, 33	sylib 218	. . . . . . . 8 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
35		fvrn0 6936	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
36	35	rgenw 3065	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
37	25	fmpt 7130	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
38	36, 37	mpbi 230	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
39		ffn 6736	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}) → (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦)
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦
41	34, 40	jctil 519	. . . . . . 7 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
42		fneq1 6659	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦))
43		fveq1 6905	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓‘𝑧) = ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧))
44	43	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
45	44	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
46	45	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
47	42, 46	anbi12d 632	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))))
48	47	spcegv 3597	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))))
49	20, 41, 48	mpsyl 68	. . . . . 6 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
50	49	exlimiv 1930	. . . . 5 ⊢ (∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
51	8, 19, 50	3syl 18	. . . 4 ⊢ (∀𝑥AC 𝑥 = V → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
52	51	alrimiv 1927	. . 3 ⊢ (∀𝑥AC 𝑥 = V → ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
53		dfac4 10162	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
54	52, 53	sylibr 234	. 2 ⊢ (∀𝑥AC 𝑥 = V → CHOICE)
55	3, 54	impbii 209	1 ⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  AC wacn 9978  CHOICEwac 10155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-card 9979  df-acn 9982  df-ac 10156

This theorem is referenced by:  dfac13  10183