Theorem nninfself 11551

Description: Domain and range of the selection function for ℕ_∞. (Contributed by Jim Kingdon, 6-Aug-2022.)

Hypothesis

Ref	Expression
nninfsel.e	⊢ 𝐸 = (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))

Assertion

Ref	Expression
nninfself	⊢ 𝐸:(2𝑜 ↑𝑚 ℕ∞)⟶ℕ∞

Distinct variable groups:   𝑖,𝑘,𝑛   𝑘,𝑞,𝑛

Allowed substitution hints:   𝐸(𝑖,𝑘,𝑛,𝑞)

Proof of Theorem nninfself

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfsel.e	. 2 ⊢ 𝐸 = (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
2		nninfsellemcl 11549	. . . . 5 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑛 ∈ ω) → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
3		eqid 2088	. . . . 5 ⊢ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
4	2, 3	fmptd 5436	. . . 4 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)):ω⟶2𝑜)
5		2onn 6260	. . . . . 6 ⊢ 2𝑜 ∈ ω
6	5	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → 2𝑜 ∈ ω)
7		omex 4398	. . . . . 6 ⊢ ω ∈ V
8	7	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → ω ∈ V)
9	6, 8	elmapd 6399	. . . 4 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜 ↑𝑚 ω) ↔ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)):ω⟶2𝑜))
10	4, 9	mpbird 165	. . 3 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜 ↑𝑚 ω))
11		nninfsellemsuc 11550	. . . . 5 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ⊆ if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
12		peano2 4400	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
13		nninfsellemcl 11549	. . . . . . 7 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ suc 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
14	12, 13	sylan2 280	. . . . . 6 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
15		suceq 4220	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑗 → suc 𝑛 = suc suc 𝑗)
16	15	raleqdv 2568	. . . . . . . 8 ⊢ (𝑛 = suc 𝑗 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
17	16	ifbid 3408	. . . . . . 7 ⊢ (𝑛 = suc 𝑗 → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
18	17, 3	fvmptg 5364	. . . . . 6 ⊢ ((suc 𝑗 ∈ ω ∧ if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
19	12, 14, 18	syl2an2 561	. . . . 5 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
20		simpr 108	. . . . . 6 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21		nninfsellemcl 11549	. . . . . 6 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
22		suceq 4220	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
23	22	raleqdv 2568	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
24	23	ifbid 3408	. . . . . . 7 ⊢ (𝑛 = 𝑗 → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
25	24, 3	fvmptg 5364	. . . . . 6 ⊢ ((𝑗 ∈ ω ∧ if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
26	20, 21, 25	syl2anc 403	. . . . 5 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
27	11, 19, 26	3sstr4d 3067	. . . 4 ⊢ ((𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
28	27	ralrimiva 2446	. . 3 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
29		fveq1 5288	. . . . . 6 ⊢ (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (𝑓‘suc 𝑗) = ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗))
30		fveq1 5288	. . . . . 6 ⊢ (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (𝑓‘𝑗) = ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
31	29, 30	sseq12d 3053	. . . . 5 ⊢ (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
32	31	ralbidv 2380	. . . 4 ⊢ (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
33		df-nninf 6770	. . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2𝑜 ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
34	32, 33	elrab2 2772	. . 3 ⊢ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ ℕ∞ ↔ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜 ↑𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
35	10, 28, 34	sylanbrc 408	. 2 ⊢ (𝑞 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ ℕ∞)
36	1, 35	fmpti 5435	1 ⊢ 𝐸:(2𝑜 ↑𝑚 ℕ∞)⟶ℕ∞

Syntax hints:   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∀wral 2359  Vcvv 2619   ⊆ wss 2997  ∅c0 3284  ifcif 3389   ↦ cmpt 3891  suc csuc 4183  ωcom 4395  ⟶wf 4998  ‘cfv 5002  (class class class)co 5634  1_𝑜c1o 6156  2_𝑜c2o 6157   ↑_𝑚 cmap 6385  ℕ_∞xnninf 6768

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770

This theorem is referenced by:  nninfsellemeq  11552  nninfsellemeqinf  11554  nninfomnilem  11556