Theorem axdclem2 10529

Description: Lemma for axdc 10530. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹‘𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)

Hypothesis

Ref	Expression
axdclem2.1	⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)

Assertion

Ref	Expression
axdclem2	⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))

Distinct variable groups:   𝑓,𝐹,𝑛   𝑦,𝐹,𝑧,𝑛   𝑓,𝑔,𝑥,𝑛   𝑔,𝑠,𝑦,𝑛   𝑧,𝑔   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑔,𝑠)

Proof of Theorem axdclem2

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8447	. . . . . . 7 ⊢ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω
2		axdclem2.1	. . . . . . . 8 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
3	2	fneq1i 6645	. . . . . . 7 ⊢ (𝐹 Fn ω ↔ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω)
4	1, 3	mpbir 230	. . . . . 6 ⊢ 𝐹 Fn ω
5	4	a1i 11	. . . . 5 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 Fn ω)
6		omex 9652	. . . . . 6 ⊢ ω ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ω ∈ V)
8	5, 7	fnexd 7224	. . . 4 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
9		fveq2 6891	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘𝑛) = (𝐹‘∅))
10		suceq 6429	. . . . . . . . 9 ⊢ (𝑛 = ∅ → suc 𝑛 = suc ∅)
11	10	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
12	9, 11	breq12d 5155	. . . . . . 7 ⊢ (𝑛 = ∅ → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
13		fveq2 6891	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
14		suceq 6429	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
15	14	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
16	13, 15	breq12d 5155	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘𝑘)𝑥(𝐹‘suc 𝑘)))
17		fveq2 6891	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘𝑛) = (𝐹‘suc 𝑘))
18		suceq 6429	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
19	18	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
20	17, 19	breq12d 5155	. . . . . . 7 ⊢ (𝑛 = suc 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
21	2	fveq1i 6892	. . . . . . . . . . . . 13 ⊢ (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
22		fr0g 8448	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
23	22	elv 3475	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
24	21, 23	eqtri 2755	. . . . . . . . . . . 12 ⊢ (𝐹‘∅) = 𝑠
25	24	breq1i 5149	. . . . . . . . . . 11 ⊢ ((𝐹‘∅)𝑥𝑧 ↔ 𝑠𝑥𝑧)
26	25	biimpri 227	. . . . . . . . . 10 ⊢ (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
27	26	eximi 1830	. . . . . . . . 9 ⊢ (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
28		peano1 7886	. . . . . . . . . 10 ⊢ ∅ ∈ ω
29	2	axdclem 10528	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (∅ ∈ ω → (𝐹‘∅)𝑥(𝐹‘suc ∅)))
30	28, 29	mpi 20	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
31	27, 30	syl3an3 1163	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
32	31	3com23 1124	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
33		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑘) ∈ V
34		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (𝐹‘suc 𝑘) ∈ V
35	33, 34	brelrn 5938	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
36		ssel 3971	. . . . . . . . . . . . 13 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘suc 𝑘) ∈ ran 𝑥 → (𝐹‘suc 𝑘) ∈ dom 𝑥))
37	35, 36	syl5 34	. . . . . . . . . . . 12 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ dom 𝑥))
38	34	eldm 5897	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑘) ∈ dom 𝑥 ↔ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧)
39	37, 38	imbitrdi 250	. . . . . . . . . . 11 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
40	39	ad2antll 728	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
41		peano2 7888	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
42	2	axdclem 10528	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (suc 𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
43	41, 42	syl5 34	. . . . . . . . . . . . 13 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
44	43	3expia 1119	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
45	44	com3r 87	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
46	45	imp 406	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
47	40, 46	syld 47	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
48	47	3adantr2 1168	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
49	48	ex 412	. . . . . . 7 ⊢ (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
50	12, 16, 20, 32, 49	finds2 7898	. . . . . 6 ⊢ (𝑛 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
51	50	com12 32	. . . . 5 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝑛 ∈ ω → (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
52	51	ralrimiv 3140	. . . 4 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∀𝑛 ∈ ω (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛))
53		fveq1 6890	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑛) = (𝐹‘𝑛))
54		fveq1 6890	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
55	53, 54	breq12d 5155	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
56	55	ralbidv 3172	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
57	8, 52, 56	spcedv 3583	. . 3 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
58	57	3exp 1117	. 2 ⊢ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))))
59		vex 3473	. . . . 5 ⊢ 𝑥 ∈ V
60	59	dmex 7909	. . . 4 ⊢ dom 𝑥 ∈ V
61	60	pwex 5374	. . 3 ⊢ 𝒫 dom 𝑥 ∈ V
62	61	ac4c 10485	. 2 ⊢ ∃𝑔∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)
63	58, 62	exlimiiv 1927	1 ⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2704   ≠ wne 2935  ∀wral 3056  Vcvv 3469   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598   class class class wbr 5142   ↦ cmpt 5225  dom cdm 5672  ran crn 5673   ↾ cres 5674  suc csuc 6365   Fn wfn 6537  ‘cfv 6542  ωcom 7862  reccrdg 8421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-ac2 10472

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-ac 10125

This theorem is referenced by:  axdc  10530