Theorem infnninf 7428

Description: The point at infinity in ℕ_∞ is the constant sequence equal to 1_o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1_o}), as fconstmpt 4802 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)

Assertion

Ref	Expression
infnninf	⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Proof of Theorem infnninf

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

Step	Hyp	Ref	Expression
1		1lt2o 6688	. . . . . 6 ⊢ 1o ∈ 2o
2	1	a1i 9	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
3	2	fmpttd 5837	. . . 4 ⊢ (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
4	3	mptru 1407	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o):ω⟶2o
5		2on 6669	. . . 4 ⊢ 2o ∈ On
6		omex 4720	. . . 4 ⊢ ω ∈ V
7		elmapg 6908	. . . 4 ⊢ ((2o ∈ On ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o))
8	5, 6, 7	mp2an 426	. . 3 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o)
9	4, 8	mpbir 146	. 2 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω)
10		peano2 4722	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11		eqidd 2235	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → 1o = 1o)
12		eqid 2234	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13		1oex 6668	. . . . . . 7 ⊢ 1o ∈ V
14	11, 12, 13	fvmpt 5759	. . . . . 6 ⊢ (suc 𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
15	10, 14	syl 14	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
16		eqidd 2235	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
17	16, 12, 13	fvmpt 5759	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
18	15, 17	eqtr4d 2270	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19		eqimss 3296	. . . 4 ⊢ (((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗) → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
20	18, 19	syl 14	. . 3 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
21	20	rgen 2597	. 2 ⊢ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22		fveq1 5674	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23		fveq1 5674	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	22, 23	sseq12d 3273	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
25	24	ralbidv 2544	. . 3 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
26		df-nninf 7424	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
27	25, 26	elrab2 2979	. 2 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ ↔ ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
28	9, 21, 27	mpbir2an 951	1 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ⊆ wss 3214   ↦ cmpt 4176  Oncon0 4489  suc csuc 4491  ωcom 4717  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  2_oc2o 6654   ↑_𝑚 cmap 6895  ℕ_∞xnninf 7423

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1o 6660  df-2o 6661  df-map 6897  df-nninf 7424

This theorem is referenced by:  nnnninf2  7431  nninfwlpoimlemdc  7481  nninfct  12765  nninffeq  16937  nnnninfen  16938