isomenndlem - Mathbox for Glauco Siliprandi

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Theorem isomenndlem 45331

Description: 𝑂 is sub-additive w.r.t. countable indexed union, implies that 𝑂 is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
isomenndlem.o	⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
isomenndlem.o0	⊢ (𝜑 → (𝑂‘∅) = 0)
isomenndlem.y	⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋)
isomenndlem.subadd	⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
isomenndlem.b	⊢ (𝜑 → 𝐵 ⊆ ℕ)
isomenndlem.f	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑌)
isomenndlem.a	⊢ 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))

**Assertion**
Ref	Expression
isomenndlem	⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^{^}‘(𝑂 ↾ 𝑌)))

Distinct variable groups: 𝐴,𝑎,𝑛 𝐵,𝑛 𝑛,𝐹 𝑂,𝑎,𝑛 𝑋,𝑎 𝑛,𝑌 𝜑,𝑎,𝑛 Allowed substitution hints: 𝐵(𝑎) 𝐹(𝑎) 𝑋(𝑛) 𝑌(𝑎)

Proof of Theorem isomenndlem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝜑 → 𝜑)
2		iftrue 4534	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐵 → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = (𝐹‘𝑛))
3	2	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = (𝐹‘𝑛))
4		isomenndlem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑌)
5		f1of 6833	. . . . . . . . . . 11 ⊢ (𝐹:𝐵–1-1-onto→𝑌 → 𝐹:𝐵⟶𝑌)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶𝑌)
7		ssun1 4172	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ (𝑌 ∪ {∅})
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ (𝑌 ∪ {∅}))
9	6, 8	fssd 6735	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐵⟶(𝑌 ∪ {∅}))
10	9	ffvelcdmda 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) ∈ (𝑌 ∪ {∅}))
11	3, 10	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ (𝑌 ∪ {∅}))
12	11	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ (𝑌 ∪ {∅}))
13		iffalse 4537	. . . . . . . . 9 ⊢ (¬ 𝑛 ∈ 𝐵 → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = ∅)
14	13	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = ∅)
15		0ex 5307	. . . . . . . . . . 11 ⊢ ∅ ∈ V
16	15	snid 4664	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
17		elun2 4177	. . . . . . . . . 10 ⊢ (∅ ∈ {∅} → ∅ ∈ (𝑌 ∪ {∅}))
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ ∅ ∈ (𝑌 ∪ {∅})
19	18	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑛 ∈ 𝐵) → ∅ ∈ (𝑌 ∪ {∅}))
20	14, 19	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ (𝑌 ∪ {∅}))
21	20	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ (𝑌 ∪ {∅}))
22	12, 21	pm2.61dan 811	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ (𝑌 ∪ {∅}))
23		isomenndlem.a	. . . . 5 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
24	22, 23	fmptd 7115	. . . 4 ⊢ (𝜑 → 𝐴:ℕ⟶(𝑌 ∪ {∅}))
25		isomenndlem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋)
26		0elpw 5354	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
27		snssi 4811	. . . . . . 7 ⊢ (∅ ∈ 𝒫 𝑋 → {∅} ⊆ 𝒫 𝑋)
28	26, 27	ax-mp 5	. . . . . 6 ⊢ {∅} ⊆ 𝒫 𝑋
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → {∅} ⊆ 𝒫 𝑋)
30	25, 29	unssd 4186	. . . 4 ⊢ (𝜑 → (𝑌 ∪ {∅}) ⊆ 𝒫 𝑋)
31	24, 30	fssd 6735	. . 3 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 𝑋)
32		nnex 12220	. . . . . 6 ⊢ ℕ ∈ V
33	32	mptex 7227	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)) ∈ V
34	23, 33	eqeltri 2829	. . . 4 ⊢ 𝐴 ∈ V
35		feq1 6698	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎:ℕ⟶𝒫 𝑋 ↔ 𝐴:ℕ⟶𝒫 𝑋))
36	35	anbi2d 629	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) ↔ (𝜑 ∧ 𝐴:ℕ⟶𝒫 𝑋)))
37		fveq1 6890	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑛) = (𝐴‘𝑛))
38	37	iuneq2d 5026	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ∪ 𝑛 ∈ ℕ (𝑎‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
39	38	fveq2d 6895	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) = (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
40		simpl 483	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 ∈ ℕ) → 𝑎 = 𝐴)
41	40	fveq1d 6893	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 ∈ ℕ) → (𝑎‘𝑛) = (𝐴‘𝑛))
42	41	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝑎‘𝑛)) = (𝑂‘(𝐴‘𝑛)))
43	42	mpteq2dva 5248	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛))))
44	43	fveq2d 6895	. . . . . 6 ⊢ (𝑎 = 𝐴 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))))
45	39, 44	breq12d 5161	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))) ↔ (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛))))))
46	36, 45	imbi12d 344	. . . 4 ⊢ (𝑎 = 𝐴 → (((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))))) ↔ ((𝜑 ∧ 𝐴:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))))))
47		isomenndlem.subadd	. . . 4 ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
48	34, 46, 47	vtocl 3549	. . 3 ⊢ ((𝜑 ∧ 𝐴:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))))
49	1, 31, 48	syl2anc 584	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))))
50	6	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → 𝐹:𝐵⟶𝑌)
51		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
52		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = ℕ → 𝐵 = ℕ)
53	52	eqcomd 2738	. . . . . . . . . . . . . 14 ⊢ (𝐵 = ℕ → ℕ = 𝐵)
54	53	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → ℕ = 𝐵)
55	51, 54	eleqtrd 2835	. . . . . . . . . . . 12 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ 𝐵)
56	55	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ 𝐵)
57	50, 56	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑌)
58		eqid 2732	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)) = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛))
59	57, 58	fmptd 7115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)):ℕ⟶𝑌)
60	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐵 = ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)))
61	55	iftrued 4536	. . . . . . . . . . . . 13 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = (𝐹‘𝑛))
62	61	mpteq2dva 5248	. . . . . . . . . . . 12 ⊢ (𝐵 = ℕ → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)) = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
63	60, 62	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝐵 = ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
64	63	feq1d 6702	. . . . . . . . . 10 ⊢ (𝐵 = ℕ → (𝐴:ℕ⟶𝑌 ↔ (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)):ℕ⟶𝑌))
65	64	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → (𝐴:ℕ⟶𝑌 ↔ (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)):ℕ⟶𝑌))
66	59, 65	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → 𝐴:ℕ⟶𝑌)
67		f1ofo 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐵–1-1-onto→𝑌 → 𝐹:𝐵–onto→𝑌)
68	4, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝐵–onto→𝑌)
69		dffo3 7103	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵–onto→𝑌 ↔ (𝐹:𝐵⟶𝑌 ∧ ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛)))
70	68, 69	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹:𝐵⟶𝑌 ∧ ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛)))
71	70	simprd 496	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛))
72	71	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛))
73		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
74		rspa 3245	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑌 ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛) ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛))
75	72, 73, 74	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛))
76	75	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛))
77		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝐵 = ℕ)
78		nfre1 3282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)
79		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
80		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → 𝐵 = ℕ)
81	79, 80	eleqtrd 2835	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ℕ)
82	81	adantll 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ℕ)
83	82	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → 𝑛 ∈ ℕ)
84	60	fveq1d 6893	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = ℕ → (𝐴‘𝑛) = ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛))
85	84	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → (𝐴‘𝑛) = ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛))
86		fvex 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑛) ∈ V
87	86, 15	ifex 4578	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V
88	87	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V)
89		eqid 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
90	89	fvmpt2 7009	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
91	81, 88, 90	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
92	2	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = (𝐹‘𝑛))
93	91, 92	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛) = (𝐹‘𝑛))
94	93	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))‘𝑛) = (𝐹‘𝑛))
95		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑛) → 𝑦 = (𝐹‘𝑛))
96	95	eqcomd 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐹‘𝑛) → (𝐹‘𝑛) = 𝑦)
97	96	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → (𝐹‘𝑛) = 𝑦)
98	85, 94, 97	3eqtrrd 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 = ℕ ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → 𝑦 = (𝐴‘𝑛))
99	98	3adant1l 1176	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → 𝑦 = (𝐴‘𝑛))
100		rspe 3246	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (𝐴‘𝑛)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
101	83, 99, 100	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑛 ∈ 𝐵 ∧ 𝑦 = (𝐹‘𝑛)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
102	101	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → (𝑛 ∈ 𝐵 → (𝑦 = (𝐹‘𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))))
103	77, 78, 102	rexlimd 3263	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → (∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
104	103	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑦 ∈ 𝑌) → (∃𝑛 ∈ 𝐵 𝑦 = (𝐹‘𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
105	76, 104	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 = ℕ) ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
106	105	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
107	66, 106	jca 512	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → (𝐴:ℕ⟶𝑌 ∧ ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
108		dffo3 7103	. . . . . . 7 ⊢ (𝐴:ℕ–onto→𝑌 ↔ (𝐴:ℕ⟶𝑌 ∧ ∀𝑦 ∈ 𝑌 ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
109	107, 108	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → 𝐴:ℕ–onto→𝑌)
110		founiiun 43963	. . . . . 6 ⊢ (𝐴:ℕ–onto→𝑌 → ∪ 𝑌 = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
111	109, 110	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = ℕ) → ∪ 𝑌 = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
112		uniun 4934	. . . . . . . 8 ⊢ ∪ (𝑌 ∪ {∅}) = (∪ 𝑌 ∪ ∪ {∅})
113	15	unisn 4930	. . . . . . . . 9 ⊢ ∪ {∅} = ∅
114	113	uneq2i 4160	. . . . . . . 8 ⊢ (∪ 𝑌 ∪ ∪ {∅}) = (∪ 𝑌 ∪ ∅)
115		un0 4390	. . . . . . . 8 ⊢ (∪ 𝑌 ∪ ∅) = ∪ 𝑌
116	112, 114, 115	3eqtrri 2765	. . . . . . 7 ⊢ ∪ 𝑌 = ∪ (𝑌 ∪ {∅})
117	116	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∪ 𝑌 = ∪ (𝑌 ∪ {∅}))
118	24	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐴:ℕ⟶(𝑌 ∪ {∅}))
119		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅)
120		isomenndlem.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
121	120	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ⊆ ℕ)
122	52	necon3bi 2967	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝐵 = ℕ → 𝐵 ≠ ℕ)
123	122	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ≠ ℕ)
124	121, 123	jca 512	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → (𝐵 ⊆ ℕ ∧ 𝐵 ≠ ℕ))
125		df-pss 3967	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊊ ℕ ↔ (𝐵 ⊆ ℕ ∧ 𝐵 ≠ ℕ))
126	124, 125	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ⊊ ℕ)
127		pssnel 4470	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ⊊ ℕ → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵))
128	126, 127	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵))
129	128	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵))
130		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → 𝑛 ∈ ℕ)
131		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → 𝑛 ∈ ℕ)
132	87	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V)
133	23	fvmpt2 7009	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V) → (𝐴‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
134	131, 132, 133	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → (𝐴‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
135	134	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → (𝐴‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
136	13	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = ∅)
137		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → 𝑦 = ∅)
138	137	eqcomd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → ∅ = 𝑦)
139	138	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → ∅ = 𝑦)
140	135, 136, 139	3eqtrrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → 𝑦 = (𝐴‘𝑛))
141	130, 140, 100	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
142	141	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
143	142	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐵) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
144	119, 78, 129, 143	exlimimdd 2212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
145	144	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
146		simplll 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → 𝜑)
147		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ (𝑌 ∪ {∅}))
148		elsni 4645	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
149	148	con3i 154	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑦 = ∅ → ¬ 𝑦 ∈ {∅})
150	149	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → ¬ 𝑦 ∈ {∅})
151		elunnel2 4150	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 ∈ {∅}) → 𝑦 ∈ 𝑌)
152	147, 150, 151	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ 𝑌)
153	152	adantll 712	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ 𝑌)
154	68	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐹:𝐵–onto→𝑌)
155		foelcdmi 6953	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐵–onto→𝑌 ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ 𝐵 (𝐹‘𝑛) = 𝑦)
156	154, 73, 155	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ 𝐵 (𝐹‘𝑛) = 𝑦)
157		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑦 ∈ 𝑌)
158	120	sselda 3982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ℕ)
159	158	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵 ∧ (𝐹‘𝑛) = 𝑦) → 𝑛 ∈ ℕ)
160	158, 87, 133	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → (𝐴‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
161	160, 3	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → (𝐴‘𝑛) = (𝐹‘𝑛))
162	161	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵 ∧ (𝐹‘𝑛) = 𝑦) → (𝐴‘𝑛) = (𝐹‘𝑛))
163		simp3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵 ∧ (𝐹‘𝑛) = 𝑦) → (𝐹‘𝑛) = 𝑦)
164	162, 163	eqtr2d 2773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵 ∧ (𝐹‘𝑛) = 𝑦) → 𝑦 = (𝐴‘𝑛))
165	159, 164, 100	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵 ∧ (𝐹‘𝑛) = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
166	165	3exp 1119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ 𝐵 → ((𝐹‘𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))))
167	166	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑛 ∈ 𝐵 → ((𝐹‘𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))))
168	157, 78, 167	rexlimd 3263	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (∃𝑛 ∈ 𝐵 (𝐹‘𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
169	156, 168	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
170	146, 153, 169	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
171	145, 170	pm2.61dan 811	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
172	171	ralrimiva 3146	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛))
173	118, 172	jca 512	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → (𝐴:ℕ⟶(𝑌 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
174		dffo3 7103	. . . . . . . 8 ⊢ (𝐴:ℕ–onto→(𝑌 ∪ {∅}) ↔ (𝐴:ℕ⟶(𝑌 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴‘𝑛)))
175	173, 174	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐴:ℕ–onto→(𝑌 ∪ {∅}))
176		founiiun 43963	. . . . . . 7 ⊢ (𝐴:ℕ–onto→(𝑌 ∪ {∅}) → ∪ (𝑌 ∪ {∅}) = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
177	175, 176	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∪ (𝑌 ∪ {∅}) = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
178	117, 177	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∪ 𝑌 = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
179	111, 178	pm2.61dan 811	. . . 4 ⊢ (𝜑 → ∪ 𝑌 = ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
180	179	fveq2d 6895	. . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑌) = (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
181		uncom 4153	. . . . . . . . 9 ⊢ ((ℕ ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (ℕ ∖ 𝐵))
182	181	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((ℕ ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (ℕ ∖ 𝐵)))
183		undif 4481	. . . . . . . . 9 ⊢ (𝐵 ⊆ ℕ ↔ (𝐵 ∪ (ℕ ∖ 𝐵)) = ℕ)
184	120, 183	sylib 217	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∪ (ℕ ∖ 𝐵)) = ℕ)
185	182, 184	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → ((ℕ ∖ 𝐵) ∪ 𝐵) = ℕ)
186	185	eqcomd 2738	. . . . . 6 ⊢ (𝜑 → ℕ = ((ℕ ∖ 𝐵) ∪ 𝐵))
187	186	mpteq1d 5243	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛))) = (𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴‘𝑛))))
188	187	fveq2d 6895	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))))
189		nfv 1917	. . . . 5 ⊢ Ⅎ𝑛𝜑
190		difexg 5327	. . . . . . 7 ⊢ (ℕ ∈ V → (ℕ ∖ 𝐵) ∈ V)
191	32, 190	ax-mp 5	. . . . . 6 ⊢ (ℕ ∖ 𝐵) ∈ V
192	191	a1i 11	. . . . 5 ⊢ (𝜑 → (ℕ ∖ 𝐵) ∈ V)
193	32	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
194	193, 120	ssexd 5324	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
195		disjdifr 4472	. . . . . 6 ⊢ ((ℕ ∖ 𝐵) ∩ 𝐵) = ∅
196	195	a1i 11	. . . . 5 ⊢ (𝜑 → ((ℕ ∖ 𝐵) ∩ 𝐵) = ∅)
197		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → 𝜑)
198		eldifi 4126	. . . . . . 7 ⊢ (𝑛 ∈ (ℕ ∖ 𝐵) → 𝑛 ∈ ℕ)
199	198	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → 𝑛 ∈ ℕ)
200		isomenndlem.o	. . . . . . . 8 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
201	200	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
202	31	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 𝑋)
203	201, 202	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝐴‘𝑛)) ∈ (0[,]+∞))
204	197, 199, 203	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴‘𝑛)) ∈ (0[,]+∞))
205	158, 203	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → (𝑂‘(𝐴‘𝑛)) ∈ (0[,]+∞))
206	189, 192, 194, 196, 204, 205	sge0splitmpt 45212	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) = ((Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))))))
207		eqid 2732	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))) = (𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))
208	205, 207	fmptd 7115	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))):𝐵⟶(0[,]+∞))
209	194, 208	sge0xrcl 45186	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))) ∈ ℝ^*)
210	209	xaddlidd 44114	. . . . 5 ⊢ (𝜑 → (0 +_𝑒 (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))))) = (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))))
211	87	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) ∈ V)
212	199, 211, 133	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝐴‘𝑛) = if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅))
213		eldifn 4127	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ 𝐵) → ¬ 𝑛 ∈ 𝐵)
214	213	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → ¬ 𝑛 ∈ 𝐵)
215	214	iffalsed 4539	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅) = ∅)
216	212, 215	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝐴‘𝑛) = ∅)
217	216	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴‘𝑛)) = (𝑂‘∅))
218		isomenndlem.o0	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘∅) = 0)
219	197, 218	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘∅) = 0)
220	217, 219	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴‘𝑛)) = 0)
221	220	mpteq2dva 5248	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛))) = (𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0))
222	221	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0)))
223	189, 192	sge0z 45176	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0)) = 0)
224	222, 223	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) = 0)
225	224	oveq1d 7426	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))))) = (0 +_𝑒 (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))))))
226	200, 25	feqresmpt 6961	. . . . . . 7 ⊢ (𝜑 → (𝑂 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (𝑂‘𝑦)))
227	226	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ 𝑌)) = (Σ^{^}‘(𝑦 ∈ 𝑌 ↦ (𝑂‘𝑦))))
228		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
229		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = (𝐴‘𝑛) → (𝑂‘𝑦) = (𝑂‘(𝐴‘𝑛)))
230	161	eqcomd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = (𝐴‘𝑛))
231	200	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
232	25	sselda 3982	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋)
233	231, 232	ffvelcdmd 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑂‘𝑦) ∈ (0[,]+∞))
234	228, 189, 229, 194, 4, 230, 233	sge0f1o 45183	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ 𝑌 ↦ (𝑂‘𝑦))) = (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))))
235		eqidd 2733	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))))
236	227, 234, 235	3eqtrd 2776	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ 𝑌)) = (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛)))))
237	210, 225, 236	3eqtr4d 2782	. . . 4 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴‘𝑛)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ 𝐵 ↦ (𝑂‘(𝐴‘𝑛))))) = (Σ^{^}‘(𝑂 ↾ 𝑌)))
238	188, 206, 237	3eqtrrd 2777	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ 𝑌)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛)))))
239	180, 238	breq12d 5161	. 2 ⊢ (𝜑 → ((𝑂‘∪ 𝑌) ≤ (Σ^{^}‘(𝑂 ↾ 𝑌)) ↔ (𝑂‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴‘𝑛))))))
240	49, 239	mpbird 256	1 ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^{^}‘(𝑂 ↾ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ⊊ wpss 3949 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5678 ⟶wf 6539 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 0cc0 11112 +∞cpnf 11247 ≤ cle 11251 ℕcn 12214 +_𝑒 cxad 13092 [,]cicc 13329 Σ^{^}csumge0 45163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-xadd 13095 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-sumge0 45164

This theorem is referenced by: isomennd 45332

W3C validator