Theorem infnninf 7022

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 6321	. . . . . 6 ⊢ 1o ∈ V
2	1	sucid 4339	. . . . 5 ⊢ 1o ∈ suc 1o
3		df-2o 6314	. . . . 5 ⊢ 2o = suc 1o
4	2, 3	eleqtrri 2215	. . . 4 ⊢ 1o ∈ 2o
5	4	fconst6 5322	. . 3 ⊢ (ω × {1o}):ω⟶2o
6		2onn 6417	. . . . 5 ⊢ 2o ∈ ω
7	6	elexi 2698	. . . 4 ⊢ 2o ∈ V
8		omex 4507	. . . 4 ⊢ ω ∈ V
9	7, 8	elmap 6571	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
10	5, 9	mpbir 145	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
11		peano2 4509	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
12	1	fvconst2 5636	. . . . . 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 5636	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
15	13, 14	eqtr4d 2175	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
16		eqimss 3151	. . . 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 2485	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
19		fveq1 5420	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
20		fveq1 5420	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
21	19, 20	sseq12d 3128	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
22	21	ralbidv 2437	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23		df-nninf 7007	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
24	22, 23	elrab2 2843	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
25	10, 18, 24	mpbir2an 926	1 ⊢ (ω × {1o}) ∈ ℕ∞

Syntax hints:   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  {csn 3527  suc csuc 4287  ωcom 4504   × cxp 4537  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  2_oc2o 6307   ↑_𝑚 cmap 6542  ℕ_∞xnninf 7005

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-nninf 7007

This theorem is referenced by:  fxnn0nninf  10211  nninffeq  13216