Theorem axdclem2 10408

Description: Lemma for axdc 10409. 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 8354	. . . . . . 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 6578	. . . . . . 7 ⊢ (𝐹 Fn ω ↔ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω)
4	1, 3	mpbir 231	. . . . . 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 9533	. . . . . 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 7152	. . . 4 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
9		fveq2 6822	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘𝑛) = (𝐹‘∅))
10		suceq 6374	. . . . . . . . 9 ⊢ (𝑛 = ∅ → suc 𝑛 = suc ∅)
11	10	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
12	9, 11	breq12d 5104	. . . . . . 7 ⊢ (𝑛 = ∅ → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
13		fveq2 6822	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
14		suceq 6374	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
15	14	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
16	13, 15	breq12d 5104	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘𝑘)𝑥(𝐹‘suc 𝑘)))
17		fveq2 6822	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘𝑛) = (𝐹‘suc 𝑘))
18		suceq 6374	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
19	18	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
20	17, 19	breq12d 5104	. . . . . . 7 ⊢ (𝑛 = suc 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
21	2	fveq1i 6823	. . . . . . . . . . . . 13 ⊢ (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
22		fr0g 8355	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
23	22	elv 3441	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
24	21, 23	eqtri 2754	. . . . . . . . . . . 12 ⊢ (𝐹‘∅) = 𝑠
25	24	breq1i 5098	. . . . . . . . . . 11 ⊢ ((𝐹‘∅)𝑥𝑧 ↔ 𝑠𝑥𝑧)
26	25	biimpri 228	. . . . . . . . . 10 ⊢ (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
27	26	eximi 1836	. . . . . . . . 9 ⊢ (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
28		peano1 7819	. . . . . . . . . 10 ⊢ ∅ ∈ ω
29	2	axdclem 10407	. . . . . . . . . 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 1165	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
32	31	3com23 1126	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
33		fvex 6835	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑘) ∈ V
34		fvex 6835	. . . . . . . . . . . . . 14 ⊢ (𝐹‘suc 𝑘) ∈ V
35	33, 34	brelrn 5882	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
36		ssel 3928	. . . . . . . . . . . . 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 5840	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑘) ∈ dom 𝑥 ↔ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧)
39	37, 38	imbitrdi 251	. . . . . . . . . . 11 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
40	39	ad2antll 729	. . . . . . . . . 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 7820	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
42	2	axdclem 10407	. . . . . . . . . . . . . 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 1121	. . . . . . . . . . . 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 1171	. . . . . . . 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 7828	. . . . . 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 3123	. . . 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 6821	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑛) = (𝐹‘𝑛))
54		fveq1 6821	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
55	53, 54	breq12d 5104	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
56	55	ralbidv 3155	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
57	8, 52, 56	spcedv 3553	. . 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 1119	. 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 3440	. . . . 5 ⊢ 𝑥 ∈ V
60	59	dmex 7839	. . . 4 ⊢ dom 𝑥 ∈ V
61	60	pwex 5318	. . 3 ⊢ 𝒫 dom 𝑥 ∈ V
62	61	ac4c 10364	. 2 ⊢ ∃𝑔∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)
63	58, 62	exlimiiv 1932	1 ⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709   ≠ wne 2928  ∀wral 3047  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550   class class class wbr 5091   ↦ cmpt 5172  dom cdm 5616  ran crn 5617   ↾ cres 5618  suc csuc 6308   Fn wfn 6476  ‘cfv 6481  ωcom 7796  reccrdg 8328

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-ac2 10351

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-ac 10004

This theorem is referenced by:  axdc  10409