Theorem dfacacn 9567

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 9566	. . . 4 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V)
2	1	elvd 3500	. . 3 ⊢ (CHOICE → AC 𝑥 = V)
3	2	alrimiv 1928	. 2 ⊢ (CHOICE → ∀𝑥AC 𝑥 = V)
4		vex 3497	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	difexi 5232	. . . . . 6 ⊢ (𝑦 ∖ {∅}) ∈ V
6		acneq 9469	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
7	6	eqeq1d 2823	. . . . . 6 ⊢ (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
8	5, 7	spcv 3606	. . . . 5 ⊢ (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
9		vuniex 7465	. . . . . . 7 ⊢ ∪ 𝑦 ∈ V
10		id 22	. . . . . . 7 ⊢ (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
11	9, 10	eleqtrrid 2920	. . . . . 6 ⊢ (AC (𝑦 ∖ {∅}) = V → ∪ 𝑦 ∈ AC (𝑦 ∖ {∅}))
12		eldifi 4103	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ∈ 𝑦)
13		elssuni 4868	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ⊆ ∪ 𝑦)
15		eldifsni 4722	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
16	14, 15	jca 514	. . . . . . 7 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅))
17	16	rgen 3148	. . . . . 6 ⊢ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)
18		acni2 9472	. . . . . 6 ⊢ ((∪ 𝑦 ∈ AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
19	11, 17, 18	sylancl 588	. . . . 5 ⊢ (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
20	4	mptex 6986	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V
21		simpr 487	. . . . . . . . 9 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧)
22		eldifsn 4719	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅))
23	22	imbi1i 352	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
24		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
25		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))
26		fvex 6683	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑧) ∈ V
27	24, 25, 26	fvmpt 6768	. . . . . . . . . . . . . 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 2897	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔‘𝑧) ∈ 𝑧))
30	29	pm5.74i 273	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧))
31		impexp 453	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
32	23, 30, 31	3bitr3i 303	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
33	32	ralbii2 3163	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
34	21, 33	sylib 220	. . . . . . . 8 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
35		fvrn0 6698	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
36	35	rgenw 3150	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
37	25	fmpt 6874	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
38	36, 37	mpbi 232	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
39		ffn 6514	. . . . . . . . 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 522	. . . . . . 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 6444	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦))
43		fveq1 6669	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓‘𝑧) = ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧))
44	43	eleq1d 2897	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
45	44	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
46	45	ralbidv 3197	. . . . . . . . 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 1931	. . . . 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 1928	. . 3 ⊢ (∀𝑥AC 𝑥 = V → ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
53		dfac4 9548	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
54	52, 53	sylibr 236	. 2 ⊢ (∀𝑥AC 𝑥 = V → CHOICE)
55	3, 54	impbii 211	1 ⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ cuni 4838   ↦ cmpt 5146  ran crn 5556   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  AC wacn 9367  CHOICEwac 9541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-card 9368  df-acn 9371  df-ac 9542

This theorem is referenced by:  dfac13  9568