Theorem infnninfOLD 7241

Description: Obsolete version of infnninf 7240 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 6540	. . . 4 ⊢ 1o ∈ 2o
2	1	fconst6 5486	. . 3 ⊢ (ω × {1o}):ω⟶2o
3		2onn 6619	. . . . 5 ⊢ 2o ∈ ω
4	3	elexi 2786	. . . 4 ⊢ 2o ∈ V
5		omex 4648	. . . 4 ⊢ ω ∈ V
6	4, 5	elmap 6776	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
7	2, 6	mpbir 146	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
8		peano2 4650	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9		1oex 6522	. . . . . . 7 ⊢ 1o ∈ V
10	9	fvconst2 5812	. . . . . 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 5812	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
13	11, 12	eqtr4d 2242	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14		eqimss 3251	. . . 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 2560	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17		fveq1 5587	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18		fveq1 5587	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
19	17, 18	sseq12d 3228	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
20	19	ralbidv 2507	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21		df-nninf 7236	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
22	20, 21	elrab2 2936	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23	7, 16, 22	mpbir2an 945	1 ⊢ (ω × {1o}) ∈ ℕ∞

Syntax hints:   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ⊆ wss 3170  {csn 3637  suc csuc 4419  ωcom 4645   × cxp 4680  ⟶wf 5275  ‘cfv 5279  (class class class)co 5956  1_oc1o 6507  2_oc2o 6508   ↑_𝑚 cmap 6747  ℕ_∞xnninf 7235

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1o 6514  df-2o 6515  df-map 6749  df-nninf 7236

This theorem is referenced by:  fxnn0nninf  10601