Theorem infnninfOLD 7089

Description: Obsolete version of infnninf 7088 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 6410	. . . 4 ⊢ 1o ∈ 2o
2	1	fconst6 5387	. . 3 ⊢ (ω × {1o}):ω⟶2o
3		2onn 6489	. . . . 5 ⊢ 2o ∈ ω
4	3	elexi 2738	. . . 4 ⊢ 2o ∈ V
5		omex 4570	. . . 4 ⊢ ω ∈ V
6	4, 5	elmap 6643	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
7	2, 6	mpbir 145	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
8		peano2 4572	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9		1oex 6392	. . . . . . 7 ⊢ 1o ∈ V
10	9	fvconst2 5701	. . . . . 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 5701	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
13	11, 12	eqtr4d 2201	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14		eqimss 3196	. . . 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 2519	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17		fveq1 5485	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18		fveq1 5485	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
19	17, 18	sseq12d 3173	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
20	19	ralbidv 2466	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21		df-nninf 7085	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
22	20, 21	elrab2 2885	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23	7, 16, 22	mpbir2an 932	1 ⊢ (ω × {1o}) ∈ ℕ∞

Syntax hints:   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  {csn 3576  suc csuc 4343  ωcom 4567   × cxp 4602  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  2_oc2o 6378   ↑_𝑚 cmap 6614  ℕ_∞xnninf 7084

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1o 6384  df-2o 6385  df-map 6616  df-nninf 7085

This theorem is referenced by:  fxnn0nninf  10373