Theorem nninfself 14847

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 14845	. . . . 5 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑛 ∈ ω) → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
3		eqid 2177	. . . . 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 5672	. . . 4 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)):ω⟶2o)
5		2onn 6524	. . . . . 6 ⊢ 2o ∈ ω
6	5	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → 2o ∈ ω)
7		omex 4594	. . . . . 6 ⊢ ω ∈ V
8	7	a1i 9	. . . . 5 ⊢ (𝑞 ∈ (2o ↑𝑚 ℕ∞) → ω ∈ V)
9	6, 8	elmapd 6664	. . . 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 167	. . 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 14846	. . . . 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 4596	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
13		nninfsellemcl 14845	. . . . . . 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 286	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
15		suceq 4404	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑗 → suc 𝑛 = suc suc 𝑗)
16	15	raleqdv 2679	. . . . . . . 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 3557	. . . . . . 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 5594	. . . . . 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 594	. . . . 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 110	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21		nninfsellemcl 14845	. . . . . 6 ⊢ ((𝑞 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
22		suceq 4404	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
23	22	raleqdv 2679	. . . . . . . 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 3557	. . . . . . 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 5594	. . . . . 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 411	. . . . 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 3202	. . . 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 2550	. . 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 5516	. . . . . 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 5516	. . . . . 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 3188	. . . . 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 2477	. . . 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 7121	. . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
34	32, 33	elrab2 2898	. . 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 417	. 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 5670	1 ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739   ⊆ wss 3131  ∅c0 3424  ifcif 3536   ↦ cmpt 4066  suc csuc 4367  ωcom 4591  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  1_oc1o 6412  2_oc2o 6413   ↑_𝑚 cmap 6650  ℕ_∞xnninf 7120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1o 6419  df-2o 6420  df-map 6652  df-nninf 7121

This theorem is referenced by:  nninfsellemeq  14848  nninfsellemeqinf  14850  nninfomnilem  14852