Theorem axdclem2 10637

Description: Lemma for axdc 10638. 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 8481	. . . . . . 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 6666	. . . . . . 7 ⊢ (𝐹 Fn ω ↔ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω)
4	1, 3	mpbir 234	. . . . . 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 9681	. . . . . 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 7255	. . . 4 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
9		fveq2 6916	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘𝑛) = (𝐹‘∅))
10		suceq 6457	. . . . . . . . 9 ⊢ (𝑛 = ∅ → suc 𝑛 = suc ∅)
11	10	fveq2d 6920	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
12	9, 11	breq12d 5121	. . . . . . 7 ⊢ (𝑛 = ∅ → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
13		fveq2 6916	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
14		suceq 6457	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
15	14	fveq2d 6920	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
16	13, 15	breq12d 5121	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘𝑘)𝑥(𝐹‘suc 𝑘)))
17		fveq2 6916	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘𝑛) = (𝐹‘suc 𝑘))
18		suceq 6457	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
19	18	fveq2d 6920	. . . . . . . 8 ⊢ (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
20	17, 19	breq12d 5121	. . . . . . 7 ⊢ (𝑛 = suc 𝑘 → ((𝐹‘𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
21	2	fveq1i 6917	. . . . . . . . . . . . 13 ⊢ (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
22		fr0g 8482	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
23	22	elv 3459	. . . . . . . . . . . . 13 ⊢ ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
24	21, 23	eqtri 2786	. . . . . . . . . . . 12 ⊢ (𝐹‘∅) = 𝑠
25	24	breq1i 5115	. . . . . . . . . . 11 ⊢ ((𝐹‘∅)𝑥𝑧 ↔ 𝑠𝑥𝑧)
26	25	biimpri 231	. . . . . . . . . 10 ⊢ (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
27	26	eximi 1868	. . . . . . . . 9 ⊢ (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
28		peano1 7934	. . . . . . . . . 10 ⊢ ∅ ∈ ω
29	2	axdclem 10636	. . . . . . . . . 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 21	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
31	27, 30	syl3an3 1183	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
32	31	3com23 1144	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
33		fvex 6929	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑘) ∈ V
34		fvex 6929	. . . . . . . . . . . . . 14 ⊢ (𝐹‘suc 𝑘) ∈ V
35	33, 34	brelrn 5936	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
36		ssel 3907	. . . . . . . . . . . . 13 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘suc 𝑘) ∈ ran 𝑥 → (𝐹‘suc 𝑘) ∈ dom 𝑥))
37	35, 36	syl5 35	. . . . . . . . . . . 12 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ dom 𝑥))
38	34	eldm 5894	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑘) ∈ dom 𝑥 ↔ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧)
39	37, 38	imbitrdi 254	. . . . . . . . . . 11 ⊢ (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
40	39	ad2antll 742	. . . . . . . . . 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 7935	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
42	2	axdclem 10636	. . . . . . . . . . . . . 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 35	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 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 88	. . . . . . . . . . 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 412	. . . . . . . . . 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 48	. . . . . . . . 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 1189	. . . . . . . 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 418	. . . . . . 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 7944	. . . . . 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 33	. . . . 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 3157	. . . 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 6915	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑛) = (𝐹‘𝑛))
54		fveq1 6915	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
55	53, 54	breq12d 5121	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
56	55	ralbidv 3189	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛)𝑥(𝐹‘suc 𝑛)))
57	8, 52, 56	spcedv 3556	. . 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 1137	. 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 3458	. . . . 5 ⊢ 𝑥 ∈ V
60	59	dmex 7955	. . . 4 ⊢ dom 𝑥 ∈ V
61	60	pwex 5349	. . 3 ⊢ 𝒫 dom 𝑥 ∈ V
62	61	ac4c 10593	. 2 ⊢ ∃𝑔∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)
63	58, 62	exlimiiv 1966	1 ⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660  suc csuc 6390   Fn wfn 6561  ‘cfv 6566  ωcom 7911  reccrdg 8455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-ac2 10580

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-ac 10234

This theorem is used by:  axdc  10638