Theorem infnninf 7030

Description: The point at infinity in ℕ_∞ (the constant sequence equal to 1_o). (Contributed by Jim Kingdon, 14-Jul-2022.)

Assertion

Ref	Expression
infnninf	⊢ (ω × {1o}) ∈ ℕ∞

Proof of Theorem infnninf

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

Step	Hyp	Ref	Expression
1		1oex 6329	. . . . . 6 ⊢ 1o ∈ V
2	1	sucid 4347	. . . . 5 ⊢ 1o ∈ suc 1o
3		df-2o 6322	. . . . 5 ⊢ 2o = suc 1o
4	2, 3	eleqtrri 2216	. . . 4 ⊢ 1o ∈ 2o
5	4	fconst6 5330	. . 3 ⊢ (ω × {1o}):ω⟶2o
6		2onn 6425	. . . . 5 ⊢ 2o ∈ ω
7	6	elexi 2701	. . . 4 ⊢ 2o ∈ V
8		omex 4515	. . . 4 ⊢ ω ∈ V
9	7, 8	elmap 6579	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
10	5, 9	mpbir 145	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
11		peano2 4517	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
12	1	fvconst2 5644	. . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
13	11, 12	syl 14	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
14	1	fvconst2 5644	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
15	13, 14	eqtr4d 2176	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
16		eqimss 3156	. . . 4 ⊢ (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
17	15, 16	syl 14	. . 3 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
18	17	rgen 2488	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
19		fveq1 5428	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
20		fveq1 5428	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
21	19, 20	sseq12d 3133	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
22	21	ralbidv 2438	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23		df-nninf 7015	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
24	22, 23	elrab2 2847	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
25	10, 18, 24	mpbir2an 927	1 ⊢ (ω × {1o}) ∈ ℕ∞

Syntax hints:   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ⊆ wss 3076  {csn 3532  suc csuc 4295  ωcom 4512   × cxp 4545  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782  1_oc1o 6314  2_oc2o 6315   ↑_𝑚 cmap 6550  ℕ_∞xnninf 7013

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1o 6321  df-2o 6322  df-map 6552  df-nninf 7015

This theorem is referenced by:  fxnn0nninf  10242  nninffeq  13391