Theorem nninfself 14046

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 14044	. . . . 5 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑛 ∈ ω) → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
3		eqid 2170	. . . . 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 5650	. . . 4 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)):ω⟶2o)
5		2onn 6500	. . . . . 6 ⊢ 2o ∈ ω
6	5	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → 2o ∈ ω)
7		omex 4577	. . . . . 6 ⊢ ω ∈ V
8	7	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → ω ∈ V)
9	6, 8	elmapd 6640	. . . 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 166	. . 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 14045	. . . . 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 4579	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
13		nninfsellemcl 14044	. . . . . . 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 284	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
15		suceq 4387	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑗 → suc 𝑛 = suc suc 𝑗)
16	15	raleqdv 2671	. . . . . . . 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 3547	. . . . . . 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 5572	. . . . . 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 589	. . . . 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 109	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21		nninfsellemcl 14044	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
22		suceq 4387	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
23	22	raleqdv 2671	. . . . . . . 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 3547	. . . . . . 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 5572	. . . . . 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 409	. . . . 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 3192	. . . 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 2543	. . 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 5495	. . . . . 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 5495	. . . . . 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 3178	. . . . 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 2470	. . . 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 7097	. . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
34	32, 33	elrab2 2889	. . 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 415	. 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 5648	1 ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∀wral 2448  Vcvv 2730   ⊆ wss 3121  ∅c0 3414  ifcif 3526   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  2_oc2o 6389   ↑_𝑚 cmap 6626  ℕ_∞xnninf 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-nninf 7097

This theorem is referenced by:  nninfsellemeq  14047  nninfsellemeqinf  14049  nninfomnilem  14051