Theorem dfacacn 10036

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 10035	. . . 4 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V)
2	1	elvd 3442	. . 3 ⊢ (CHOICE → AC 𝑥 = V)
3	2	alrimiv 1927	. 2 ⊢ (CHOICE → ∀𝑥AC 𝑥 = V)
4		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	difexi 5269	. . . . . 6 ⊢ (𝑦 ∖ {∅}) ∈ V
6		acneq 9937	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
7	6	eqeq1d 2731	. . . . . 6 ⊢ (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
8	5, 7	spcv 3560	. . . . 5 ⊢ (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
9		vuniex 7675	. . . . . . 7 ⊢ ∪ 𝑦 ∈ V
10		id 22	. . . . . . 7 ⊢ (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
11	9, 10	eleqtrrid 2835	. . . . . 6 ⊢ (AC (𝑦 ∖ {∅}) = V → ∪ 𝑦 ∈ AC (𝑦 ∖ {∅}))
12		eldifi 4082	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ∈ 𝑦)
13		elssuni 4888	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ⊆ ∪ 𝑦)
15		eldifsni 4741	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
16	14, 15	jca 511	. . . . . . 7 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅))
17	16	rgen 3046	. . . . . 6 ⊢ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)
18		acni2 9940	. . . . . 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 7159	. . . . . . 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 4737	. . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
25		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))
26		fvex 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑧) ∈ V
27	24, 25, 26	fvmpt 6930	. . . . . . . . . . . . . 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 2813	. . . . . . . . . . . 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 3071	. . . . . . . . 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 6850	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
36	35	rgenw 3048	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
37	25	fmpt 7044	. . . . . . . . . 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 6652	. . . . . . . . 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 6573	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦))
43		fveq1 6821	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓‘𝑧) = ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧))
44	43	eleq1d 2813	. . . . . . . . . . 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 3152	. . . . . . . . 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 3552	. . . . . . 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 10016	. . 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 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577  ∪ cuni 4858   ↦ cmpt 5173  ran crn 5620   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  AC wacn 9834  CHOICEwac 10009

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-card 9835  df-acn 9838  df-ac 10010

This theorem is referenced by:  dfac13  10037