Theorem infnninf 6784

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 6171	. . . . . 6 ⊢ 1𝑜 ∈ V
2	1	sucid 4235	. . . . 5 ⊢ 1𝑜 ∈ suc 1𝑜
3		df-2o 6164	. . . . 5 ⊢ 2𝑜 = suc 1𝑜
4	2, 3	eleqtrri 2163	. . . 4 ⊢ 1𝑜 ∈ 2𝑜
5	4	fconst6 5194	. . 3 ⊢ (ω × {1𝑜}):ω⟶2𝑜
6		2onn 6260	. . . . 5 ⊢ 2𝑜 ∈ ω
7	6	elexi 2631	. . . 4 ⊢ 2𝑜 ∈ V
8		omex 4398	. . . 4 ⊢ ω ∈ V
9	7, 8	elmap 6414	. . 3 ⊢ ((ω × {1𝑜}) ∈ (2𝑜 ↑𝑚 ω) ↔ (ω × {1𝑜}):ω⟶2𝑜)
10	5, 9	mpbir 144	. 2 ⊢ (ω × {1𝑜}) ∈ (2𝑜 ↑𝑚 ω)
11		peano2 4400	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
12	1	fvconst2 5495	. . . . . 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 5495	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1𝑜})‘𝑖) = 1𝑜)
15	13, 14	eqtr4d 2123	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1𝑜})‘suc 𝑖) = ((ω × {1𝑜})‘𝑖))
16		eqimss 3076	. . . 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 2428	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)
19		fveq1 5288	. . . . 5 ⊢ (𝑓 = (ω × {1𝑜}) → (𝑓‘suc 𝑖) = ((ω × {1𝑜})‘suc 𝑖))
20		fveq1 5288	. . . . 5 ⊢ (𝑓 = (ω × {1𝑜}) → (𝑓‘𝑖) = ((ω × {1𝑜})‘𝑖))
21	19, 20	sseq12d 3053	. . . 4 ⊢ (𝑓 = (ω × {1𝑜}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
22	21	ralbidv 2380	. . 3 ⊢ (𝑓 = (ω × {1𝑜}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
23		df-nninf 6770	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2𝑜 ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
24	22, 23	elrab2 2772	. 2 ⊢ ((ω × {1𝑜}) ∈ ℕ∞ ↔ ((ω × {1𝑜}) ∈ (2𝑜 ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
25	10, 18, 24	mpbir2an 888	1 ⊢ (ω × {1𝑜}) ∈ ℕ∞

Syntax hints:   = wceq 1289   ∈ wcel 1438  ∀wral 2359   ⊆ wss 2997  {csn 3441  suc csuc 4183  ωcom 4395   × cxp 4426  ⟶wf 4998  ‘cfv 5002  (class class class)co 5634  1_𝑜c1o 6156  2_𝑜c2o 6157   ↑_𝑚 cmap 6385  ℕ_∞xnninf 6768

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770

This theorem is referenced by:  fxnn0nninf  9809