Theorem nnsf 16633

Description: Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)

Hypothesis

Ref	Expression
nns.s	⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))

Assertion

Ref	Expression
nnsf	⊢ 𝑆:ℕ∞⟶ℕ∞

Distinct variable group:   𝑖,𝑝

Allowed substitution hints:   𝑆(𝑖,𝑝)

Proof of Theorem nnsf

Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nns.s	. 2 ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))
2		1lt2o 6610	. . . . . . 7 ⊢ 1o ∈ 2o
3	2	a1i 9	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
4		nninff 7321	. . . . . . . 8 ⊢ (𝑝 ∈ ℕ∞ → 𝑝:ω⟶2o)
5	4	adantr 276	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → 𝑝:ω⟶2o)
6		nnpredcl 4721	. . . . . . . 8 ⊢ (𝑖 ∈ ω → ∪ 𝑖 ∈ ω)
7	6	adantl 277	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → ∪ 𝑖 ∈ ω)
8	5, 7	ffvelcdmd 5783	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → (𝑝‘∪ 𝑖) ∈ 2o)
9		nndceq0 4716	. . . . . . 7 ⊢ (𝑖 ∈ ω → DECID 𝑖 = ∅)
10	9	adantl 277	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → DECID 𝑖 = ∅)
11	3, 8, 10	ifcldcd 3643	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) ∈ 2o)
12		eqid 2231	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))
13	11, 12	fmptd 5801	. . . 4 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))):ω⟶2o)
14		2onn 6689	. . . . 5 ⊢ 2o ∈ ω
15		omex 4691	. . . . 5 ⊢ ω ∈ V
16		elmapg 6830	. . . . 5 ⊢ ((2o ∈ ω ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))):ω⟶2o))
17	14, 15, 16	mp2an 426	. . . 4 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))):ω⟶2o)
18	13, 17	sylibr 134	. . 3 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω))
19		1on 6589	. . . . . . . . 9 ⊢ 1o ∈ On
20	19	ontrci 4524	. . . . . . . 8 ⊢ Tr 1o
21	2	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 1o ∈ 2o)
22	4	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 𝑝:ω⟶2o)
23		peano2 4693	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
24	23	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25		nnpredcl 4721	. . . . . . . . . . . . 13 ⊢ (suc 𝑗 ∈ ω → ∪ suc 𝑗 ∈ ω)
26	24, 25	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ∪ suc 𝑗 ∈ ω)
27	22, 26	ffvelcdmd 5783	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑝‘∪ suc 𝑗) ∈ 2o)
28		nndceq0 4716	. . . . . . . . . . . 12 ⊢ (suc 𝑗 ∈ ω → DECID suc 𝑗 = ∅)
29	24, 28	syl 14	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → DECID suc 𝑗 = ∅)
30	21, 27, 29	ifcldcd 3643	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ 2o)
31	30	adantr 276	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ 2o)
32		df-2o 6583	. . . . . . . . 9 ⊢ 2o = suc 1o
33	31, 32	eleqtrdi 2324	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ suc 1o)
34		trsucss 4520	. . . . . . . 8 ⊢ (Tr 1o → (if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ suc 1o → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ 1o))
35	20, 33, 34	mpsyl 65	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ 1o)
36		iftrue 3610	. . . . . . . 8 ⊢ (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = 1o)
37	36	adantl 277	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = 1o)
38	35, 37	sseqtrrd 3266	. . . . . 6 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
39		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
40	39	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
41		nnord 4710	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ω → Ord 𝑗)
42		ordtr 4475	. . . . . . . . . . 11 ⊢ (Ord 𝑗 → Tr 𝑗)
43	40, 41, 42	3syl 17	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44		unisucg 4511	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ω → (Tr 𝑗 ↔ ∪ suc 𝑗 = 𝑗))
45	40, 44	syl 14	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (Tr 𝑗 ↔ ∪ suc 𝑗 = 𝑗))
46	43, 45	mpbid 147	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∪ suc 𝑗 = 𝑗)
47	46	fveq2d 5643	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘∪ suc 𝑗) = (𝑝‘𝑗))
48		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
49	48	neqned 2409	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50		nnsucpred 4715	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc ∪ 𝑗 = 𝑗)
51	40, 49, 50	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc ∪ 𝑗 = 𝑗)
52	51	fveq2d 5643	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc ∪ 𝑗) = (𝑝‘𝑗))
53		suceq 4499	. . . . . . . . . . . 12 ⊢ (𝑘 = ∪ 𝑗 → suc 𝑘 = suc ∪ 𝑗)
54	53	fveq2d 5643	. . . . . . . . . . 11 ⊢ (𝑘 = ∪ 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc ∪ 𝑗))
55		fveq2 5639	. . . . . . . . . . 11 ⊢ (𝑘 = ∪ 𝑗 → (𝑝‘𝑘) = (𝑝‘∪ 𝑗))
56	54, 55	sseq12d 3258	. . . . . . . . . 10 ⊢ (𝑘 = ∪ 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘) ↔ (𝑝‘suc ∪ 𝑗) ⊆ (𝑝‘∪ 𝑗)))
57		fveq1 5638	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58		fveq1 5638	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑝 → (𝑓‘𝑗) = (𝑝‘𝑗))
59	57, 58	sseq12d 3258	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
60	59	ralbidv 2532	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
61		df-nninf 7319	. . . . . . . . . . . . . 14 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
62	60, 61	elrab2 2965	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ∞ ↔ (𝑝 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
63	62	simprbi 275	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗))
64		suceq 4499	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
65	64	fveq2d 5643	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66		fveq2 5639	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑝‘𝑗) = (𝑝‘𝑘))
67	65, 66	sseq12d 3258	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘)))
68	67	cbvralv 2767	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗) ↔ ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
69	63, 68	sylib 122	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
70	69	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
71		nnpredcl 4721	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ω → ∪ 𝑗 ∈ ω)
72	71	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ∪ 𝑗 ∈ ω)
73	72	adantr 276	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∪ 𝑗 ∈ ω)
74	56, 70, 73	rspcdva 2915	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc ∪ 𝑗) ⊆ (𝑝‘∪ 𝑗))
75	52, 74	eqsstrrd 3264	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘𝑗) ⊆ (𝑝‘∪ 𝑗))
76	47, 75	eqsstrd 3263	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘∪ suc 𝑗) ⊆ (𝑝‘∪ 𝑗))
77		peano3 4694	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
78	77	neneqd 2423	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
79	78	ad2antlr 489	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
80	79	iffalsed 3615	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) = (𝑝‘∪ suc 𝑗))
81	48	iffalsed 3615	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = (𝑝‘∪ 𝑗))
82	76, 80, 81	3sstr4d 3272	. . . . . 6 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
83		nndceq0 4716	. . . . . . . 8 ⊢ (𝑗 ∈ ω → DECID 𝑗 = ∅)
84	83	adantl 277	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → DECID 𝑗 = ∅)
85		exmiddc 843	. . . . . . 7 ⊢ (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
86	84, 85	syl 14	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
87	38, 82, 86	mpjaodan 805	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
88		eqeq1 2238	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89		unieq 3902	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → ∪ 𝑖 = ∪ suc 𝑗)
90	89	fveq2d 5643	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (𝑝‘∪ 𝑖) = (𝑝‘∪ suc 𝑗))
91	88, 90	ifbieq2d 3630	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)))
92	91, 12	fvmptg 5722	. . . . . 6 ⊢ ((suc 𝑗 ∈ ω ∧ if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)))
93	24, 30, 92	syl2anc 411	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)))
94	22, 72	ffvelcdmd 5783	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑝‘∪ 𝑗) ∈ 2o)
95	21, 94, 84	ifcldcd 3643	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) ∈ 2o)
96		eqeq1 2238	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97		unieq 3902	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ∪ 𝑖 = ∪ 𝑗)
98	97	fveq2d 5643	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑝‘∪ 𝑖) = (𝑝‘∪ 𝑗))
99	96, 98	ifbieq2d 3630	. . . . . . 7 ⊢ (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
100	99, 12	fvmptg 5722	. . . . . 6 ⊢ ((𝑗 ∈ ω ∧ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
101	39, 95, 100	syl2anc 411	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
102	87, 93, 101	3sstr4d 3272	. . . 4 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗))
103	102	ralrimiva 2605	. . 3 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗))
104		fveq1 5638	. . . . . 6 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗))
105		fveq1 5638	. . . . . 6 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗))
106	104, 105	sseq12d 3258	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗)))
107	106	ralbidv 2532	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗)))
108	107, 61	elrab2 2965	. . 3 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ ℕ∞ ↔ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗)))
109	18, 103, 108	sylanbrc 417	. 2 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ ℕ∞)
110	1, 109	fmpti 5799	1 ⊢ 𝑆:ℕ∞⟶ℕ∞

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∀wral 2510  Vcvv 2802   ⊆ wss 3200  ∅c0 3494  ifcif 3605  ∪ cuni 3893   ↦ cmpt 4150  Tr wtr 4187  Ord word 4459  suc csuc 4462  ωcom 4688  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  1_oc1o 6575  2_oc2o 6576   ↑_𝑚 cmap 6817  ℕ_∞xnninf 7318

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1o 6582  df-2o 6583  df-map 6819  df-nninf 7319

This theorem is referenced by:  peano4nninf  16634