Theorem nnsf 14038

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 6421	. . . . . . 7 ⊢ 1o ∈ 2o
3	2	a1i 9	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
4		nninff 7099	. . . . . . . 8 ⊢ (𝑝 ∈ ℕ∞ → 𝑝:ω⟶2o)
5	4	adantr 274	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → 𝑝:ω⟶2o)
6		nnpredcl 4607	. . . . . . . 8 ⊢ (𝑖 ∈ ω → ∪ 𝑖 ∈ ω)
7	6	adantl 275	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → ∪ 𝑖 ∈ ω)
8	5, 7	ffvelrnd 5632	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → (𝑝‘∪ 𝑖) ∈ 2o)
9		nndceq0 4602	. . . . . . 7 ⊢ (𝑖 ∈ ω → DECID 𝑖 = ∅)
10	9	adantl 275	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → DECID 𝑖 = ∅)
11	3, 8, 10	ifcldcd 3561	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑖 ∈ ω) → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) ∈ 2o)
12		eqid 2170	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))
13	11, 12	fmptd 5650	. . . 4 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))):ω⟶2o)
14		2onn 6500	. . . . 5 ⊢ 2o ∈ ω
15		omex 4577	. . . . 5 ⊢ ω ∈ V
16		elmapg 6639	. . . . 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 424	. . . 4 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))):ω⟶2o)
18	13, 17	sylibr 133	. . 3 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ (2o ↑𝑚 ω))
19		1on 6402	. . . . . . . . 9 ⊢ 1o ∈ On
20	19	ontrci 4412	. . . . . . . 8 ⊢ Tr 1o
21	2	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 1o ∈ 2o)
22	4	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 𝑝:ω⟶2o)
23		peano2 4579	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
24	23	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → suc 𝑗 ∈ ω)
25		nnpredcl 4607	. . . . . . . . . . . . 13 ⊢ (suc 𝑗 ∈ ω → ∪ suc 𝑗 ∈ ω)
26	24, 25	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ∪ suc 𝑗 ∈ ω)
27	22, 26	ffvelrnd 5632	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑝‘∪ suc 𝑗) ∈ 2o)
28		nndceq0 4602	. . . . . . . . . . . 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 3561	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ 2o)
31	30	adantr 274	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ 2o)
32		df-2o 6396	. . . . . . . . 9 ⊢ 2o = suc 1o
33	31, 32	eleqtrdi 2263	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ∈ suc 1o)
34		trsucss 4408	. . . . . . . 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 3531	. . . . . . . 8 ⊢ (𝑗 = ∅ → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = 1o)
37	36	adantl 275	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = 1o)
38	35, 37	sseqtrrd 3186	. . . . . 6 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
39		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
40	39	adantr 274	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ∈ ω)
41		nnord 4596	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ω → Ord 𝑗)
42		ordtr 4363	. . . . . . . . . . 11 ⊢ (Ord 𝑗 → Tr 𝑗)
43	40, 41, 42	3syl 17	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → Tr 𝑗)
44		unisucg 4399	. . . . . . . . . . 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 146	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∪ suc 𝑗 = 𝑗)
47	46	fveq2d 5500	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘∪ suc 𝑗) = (𝑝‘𝑗))
48		simpr 109	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ 𝑗 = ∅)
49	48	neqned 2347	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → 𝑗 ≠ ∅)
50		nnsucpred 4601	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → suc ∪ 𝑗 = 𝑗)
51	40, 49, 50	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → suc ∪ 𝑗 = 𝑗)
52	51	fveq2d 5500	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc ∪ 𝑗) = (𝑝‘𝑗))
53		suceq 4387	. . . . . . . . . . . 12 ⊢ (𝑘 = ∪ 𝑗 → suc 𝑘 = suc ∪ 𝑗)
54	53	fveq2d 5500	. . . . . . . . . . 11 ⊢ (𝑘 = ∪ 𝑗 → (𝑝‘suc 𝑘) = (𝑝‘suc ∪ 𝑗))
55		fveq2 5496	. . . . . . . . . . 11 ⊢ (𝑘 = ∪ 𝑗 → (𝑝‘𝑘) = (𝑝‘∪ 𝑗))
56	54, 55	sseq12d 3178	. . . . . . . . . 10 ⊢ (𝑘 = ∪ 𝑗 → ((𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘) ↔ (𝑝‘suc ∪ 𝑗) ⊆ (𝑝‘∪ 𝑗)))
57		fveq1 5495	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑝 → (𝑓‘suc 𝑗) = (𝑝‘suc 𝑗))
58		fveq1 5495	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑝 → (𝑓‘𝑗) = (𝑝‘𝑗))
59	57, 58	sseq12d 3178	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑝 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
60	59	ralbidv 2470	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑝 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
61		df-nninf 7097	. . . . . . . . . . . . . 14 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
62	60, 61	elrab2 2889	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ∞ ↔ (𝑝 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗)))
63	62	simprbi 273	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗))
64		suceq 4387	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
65	64	fveq2d 5500	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑝‘suc 𝑗) = (𝑝‘suc 𝑘))
66		fveq2 5496	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑝‘𝑗) = (𝑝‘𝑘))
67	65, 66	sseq12d 3178	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ((𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗) ↔ (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘)))
68	67	cbvralv 2696	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ ω (𝑝‘suc 𝑗) ⊆ (𝑝‘𝑗) ↔ ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
69	63, 68	sylib 121	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
70	69	ad2antrr 485	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∀𝑘 ∈ ω (𝑝‘suc 𝑘) ⊆ (𝑝‘𝑘))
71		nnpredcl 4607	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ω → ∪ 𝑗 ∈ ω)
72	71	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ∪ 𝑗 ∈ ω)
73	72	adantr 274	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ∪ 𝑗 ∈ ω)
74	56, 70, 73	rspcdva 2839	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘suc ∪ 𝑗) ⊆ (𝑝‘∪ 𝑗))
75	52, 74	eqsstrrd 3184	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘𝑗) ⊆ (𝑝‘∪ 𝑗))
76	47, 75	eqsstrd 3183	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → (𝑝‘∪ suc 𝑗) ⊆ (𝑝‘∪ 𝑗))
77		peano3 4580	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → suc 𝑗 ≠ ∅)
78	77	neneqd 2361	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → ¬ suc 𝑗 = ∅)
79	78	ad2antlr 486	. . . . . . . 8 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → ¬ suc 𝑗 = ∅)
80	79	iffalsed 3536	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) = (𝑝‘∪ suc 𝑗))
81	48	iffalsed 3536	. . . . . . 7 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = (𝑝‘∪ 𝑗))
82	76, 80, 81	3sstr4d 3192	. . . . . 6 ⊢ (((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 = ∅) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
83		nndceq0 4602	. . . . . . . 8 ⊢ (𝑗 ∈ ω → DECID 𝑗 = ∅)
84	83	adantl 275	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → DECID 𝑗 = ∅)
85		exmiddc 831	. . . . . . 7 ⊢ (DECID 𝑗 = ∅ → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
86	84, 85	syl 14	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑗 = ∅ ∨ ¬ 𝑗 = ∅))
87	38, 82, 86	mpjaodan 793	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)) ⊆ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
88		eqeq1 2177	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (𝑖 = ∅ ↔ suc 𝑗 = ∅))
89		unieq 3805	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → ∪ 𝑖 = ∪ suc 𝑗)
90	89	fveq2d 5500	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → (𝑝‘∪ 𝑖) = (𝑝‘∪ suc 𝑗))
91	88, 90	ifbieq2d 3550	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)))
92	91, 12	fvmptg 5572	. . . . . 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 409	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) = if(suc 𝑗 = ∅, 1o, (𝑝‘∪ suc 𝑗)))
94	22, 72	ffvelrnd 5632	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → (𝑝‘∪ 𝑗) ∈ 2o)
95	21, 94, 84	ifcldcd 3561	. . . . . 6 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) ∈ 2o)
96		eqeq1 2177	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑖 = ∅ ↔ 𝑗 = ∅))
97		unieq 3805	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ∪ 𝑖 = ∪ 𝑗)
98	97	fveq2d 5500	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑝‘∪ 𝑖) = (𝑝‘∪ 𝑗))
99	96, 98	ifbieq2d 3550	. . . . . . 7 ⊢ (𝑖 = 𝑗 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
100	99, 12	fvmptg 5572	. . . . . 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 409	. . . . 5 ⊢ ((𝑝 ∈ ℕ∞ ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗) = if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))
102	87, 93, 101	3sstr4d 3192	. . . 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 2543	. . 3 ⊢ (𝑝 ∈ ℕ∞ → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗))
104		fveq1 5495	. . . . . 6 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘suc 𝑗))
105		fveq1 5495	. . . . . 6 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))‘𝑗))
106	104, 105	sseq12d 3178	. . . . 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 2470	. . . 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 2889	. . 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 415	. 2 ⊢ (𝑝 ∈ ℕ∞ → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) ∈ ℕ∞)
110	1, 109	fmpti 5648	1 ⊢ 𝑆:ℕ∞⟶ℕ∞

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  Vcvv 2730   ⊆ wss 3121  ∅c0 3414  ifcif 3526  ∪ cuni 3796   ↦ cmpt 4050  Tr wtr 4087  Ord word 4347  suc csuc 4350  ωcom 4574  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  2_oc2o 6389   ↑_𝑚 cmap 6626  ℕ_∞xnninf 7096

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-nninf 7097

This theorem is referenced by:  peano4nninf  14039