Theorem infnninf 7088

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 4651 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 6410	. . . . . 6 ⊢ 1o ∈ 2o
2	1	a1i 9	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
3	2	fmpttd 5640	. . . 4 ⊢ (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
4	3	mptru 1352	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o):ω⟶2o
5		2on 6393	. . . 4 ⊢ 2o ∈ On
6		omex 4570	. . . 4 ⊢ ω ∈ V
7		elmapg 6627	. . . 4 ⊢ ((2o ∈ On ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o))
8	5, 6, 7	mp2an 423	. . 3 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o)
9	4, 8	mpbir 145	. 2 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω)
10		peano2 4572	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11		eqidd 2166	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → 1o = 1o)
12		eqid 2165	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13		1oex 6392	. . . . . . 7 ⊢ 1o ∈ V
14	11, 12, 13	fvmpt 5563	. . . . . 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 2166	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
17	16, 12, 13	fvmpt 5563	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
18	15, 17	eqtr4d 2201	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19		eqimss 3196	. . . 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 2519	. 2 ⊢ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22		fveq1 5485	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23		fveq1 5485	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	22, 23	sseq12d 3173	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
25	24	ralbidv 2466	. . 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 7085	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
27	25, 26	elrab2 2885	. 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 932	1 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ⊤wtru 1344   ∈ wcel 2136  ∀wral 2444  Vcvv 2726   ⊆ wss 3116   ↦ cmpt 4043  Oncon0 4341  suc csuc 4343  ωcom 4567  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  2_oc2o 6378   ↑_𝑚 cmap 6614  ℕ_∞xnninf 7084

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1o 6384  df-2o 6385  df-map 6616  df-nninf 7085

This theorem is referenced by:  nnnninf2  7091  nninffeq  13900