Theorem dfacacn 8846

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		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
2		acacni 8845	. . . 4 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V)
3	1, 2	mpan2 703	. . 3 ⊢ (CHOICE → AC 𝑥 = V)
4	3	alrimiv 1842	. 2 ⊢ (CHOICE → ∀𝑥AC 𝑥 = V)
5		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
6		difexg 4735	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∖ {∅}) ∈ V)
7	5, 6	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∖ {∅}) ∈ V
8		acneq 8749	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
9	8	eqeq1d 2612	. . . . . 6 ⊢ (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
10	7, 9	spcv 3272	. . . . 5 ⊢ (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
11		vuniex 6852	. . . . . . 7 ⊢ ∪ 𝑦 ∈ V
12		id 22	. . . . . . 7 ⊢ (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
13	11, 12	syl5eleqr 2695	. . . . . 6 ⊢ (AC (𝑦 ∖ {∅}) = V → ∪ 𝑦 ∈ AC (𝑦 ∖ {∅}))
14		eldifi 3694	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ∈ 𝑦)
15		elssuni 4403	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ⊆ ∪ 𝑦)
17		eldifsni 4261	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
18	16, 17	jca 553	. . . . . . 7 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅))
19	18	rgen 2906	. . . . . 6 ⊢ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)
20		acni2 8752	. . . . . 6 ⊢ ((∪ 𝑦 ∈ AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
21	13, 19, 20	sylancl 693	. . . . 5 ⊢ (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
22	5	mptex 6390	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V
23		simpr 476	. . . . . . . . 9 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧)
24		eldifsn 4260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅))
25	24	imbi1i 338	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
26		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
27		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))
28		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑧) ∈ V
29	26, 27, 28	fvmpt 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
30	14, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
31	30	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔‘𝑧) ∈ 𝑧))
32	31	pm5.74i 259	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧))
33		impexp 461	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
34	25, 32, 33	3bitr3i 289	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
35	34	ralbii2 2961	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
36	23, 35	sylib 207	. . . . . . . 8 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
37		fvrn0 6126	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
38	37	rgenw 2908	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
39	27	fmpt 6289	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
40	38, 39	mpbi 219	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
41		ffn 5958	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}) → (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦)
42	40, 41	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦
43	36, 42	jctil 558	. . . . . . 7 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
44		fneq1 5893	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦))
45		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓‘𝑧) = ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧))
46	45	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
47	46	imbi2d 329	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
48	47	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
49	44, 48	anbi12d 743	. . . . . . . 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))))
50	49	spcegv 3267	. . . . . . 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))))
51	22, 43, 50	mpsyl 66	. . . . . 6 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
52	51	exlimiv 1845	. . . . 5 ⊢ (∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
53	10, 21, 52	3syl 18	. . . 4 ⊢ (∀𝑥AC 𝑥 = V → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
54	53	alrimiv 1842	. . 3 ⊢ (∀𝑥AC 𝑥 = V → ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
55		dfac4 8828	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
56	54, 55	sylibr 223	. 2 ⊢ (∀𝑥AC 𝑥 = V → CHOICE)
57	4, 56	impbii 198	1 ⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  AC wacn 8647  CHOICEwac 8821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-card 8648  df-acn 8651  df-ac 8822

This theorem is referenced by:  dfac13  8847