Theorem infnninfOLD 7367

Description: Obsolete version of infnninf 7366 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem infnninfOLD

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

Step	Hyp	Ref	Expression
1		1lt2o 6653	. . . 4 ⊢ 1o ∈ 2o
2	1	fconst6 5545	. . 3 ⊢ (ω × {1o}):ω⟶2o
3		2onn 6732	. . . . 5 ⊢ 2o ∈ ω
4	3	elexi 2816	. . . 4 ⊢ 2o ∈ V
5		omex 4697	. . . 4 ⊢ ω ∈ V
6	4, 5	elmap 6889	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
7	2, 6	mpbir 146	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
8		peano2 4699	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9		1oex 6633	. . . . . . 7 ⊢ 1o ∈ V
10	9	fvconst2 5878	. . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
11	8, 10	syl 14	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
12	9	fvconst2 5878	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
13	11, 12	eqtr4d 2267	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14		eqimss 3282	. . . 4 ⊢ (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
15	13, 14	syl 14	. . 3 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
16	15	rgen 2586	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17		fveq1 5647	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18		fveq1 5647	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
19	17, 18	sseq12d 3259	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
20	19	ralbidv 2533	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21		df-nninf 7362	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
22	20, 21	elrab2 2966	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23	7, 16, 22	mpbir2an 951	1 ⊢ (ω × {1o}) ∈ ℕ∞

Syntax hints:   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ⊆ wss 3201  {csn 3673  suc csuc 4468  ωcom 4694   × cxp 4729  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1_oc1o 6618  2_oc2o 6619   ↑_𝑚 cmap 6860  ℕ_∞xnninf 7361

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-map 6862  df-nninf 7362

This theorem is referenced by:  fxnn0nninf  10747