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Theorem dyadmbl 25529

Description: Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypotheses**
Ref	Expression
dyadmbl.1	⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
dyadmbl.2	⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
dyadmbl.3	⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

**Assertion**
Ref	Expression
dyadmbl	⊢ (𝜑 → ∪ ([,] “ 𝐴) ∈ dom vol)

Distinct variable groups: 𝑥,𝑦 𝑧,𝑤,𝜑 𝑥,𝑤,𝑦,𝐴,𝑧 𝑧,𝐺 𝑤,𝐹,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐺(𝑥,𝑦,𝑤)

Proof of Theorem dyadmbl Dummy variables 𝑓 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dyadmbl.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
2		dyadmbl.2	. . 3 ⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
3		dyadmbl.3	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
4	1, 2, 3	dyadmbllem 25528	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))
5		isfinite 9542	. . . 4 ⊢ (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6		iccf 13348	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
7		ffun 6654	. . . . . 6 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
8		funiunfv 7182	. . . . . 6 ⊢ (Fun [,] → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) = ∪ ([,] “ 𝐺))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) = ∪ ([,] “ 𝐺)
10		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → 𝐺 ∈ Fin)
11	2	ssrab3 4032	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ⊆ 𝐴
12	11, 3	sstrid 3946	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ ran 𝐹)
13	1	dyadf 25520	. . . . . . . . . . . . . . . 16 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
14		frn 6658	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
15	13, 14	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
16		inss2 4188	. . . . . . . . . . . . . . 15 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
17	15, 16	sstri 3944	. . . . . . . . . . . . . 14 ⊢ ran 𝐹 ⊆ (ℝ × ℝ)
18	12, 17	sstrdi 3947	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ (ℝ × ℝ))
19	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
20	19	sselda 3934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → 𝑛 ∈ (ℝ × ℝ))
21		1st2nd2 7960	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
23	22	fveq2d 6826	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) = ([,]‘⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩))
24		df-ov 7349	. . . . . . . . 9 ⊢ ((1^st ‘𝑛)[,](2^nd ‘𝑛)) = ([,]‘⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
25	23, 24	eqtr4di 2784	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) = ((1^st ‘𝑛)[,](2^nd ‘𝑛)))
26		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℝ × ℝ) → (1^st ‘𝑛) ∈ ℝ)
27	20, 26	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → (1^st ‘𝑛) ∈ ℝ)
28		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℝ × ℝ) → (2^nd ‘𝑛) ∈ ℝ)
29	20, 28	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → (2^nd ‘𝑛) ∈ ℝ)
30		iccmbl 25495	. . . . . . . . 9 ⊢ (((1^st ‘𝑛) ∈ ℝ ∧ (2^nd ‘𝑛) ∈ ℝ) → ((1^st ‘𝑛)[,](2^nd ‘𝑛)) ∈ dom vol)
31	27, 29, 30	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ((1^st ‘𝑛)[,](2^nd ‘𝑛)) ∈ dom vol)
32	25, 31	eqeltrd 2831	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) ∈ dom vol)
33	32	ralrimiva 3124	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∀𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
34		finiunmbl 25473	. . . . . 6 ⊢ ((𝐺 ∈ Fin ∧ ∀𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol) → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
35	10, 33, 34	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
36	9, 35	eqeltrrid 2836	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∪ ([,] “ 𝐺) ∈ dom vol)
37	5, 36	sylan2br 595	. . 3 ⊢ ((𝜑 ∧ 𝐺 ≺ ω) → ∪ ([,] “ 𝐺) ∈ dom vol)
38		rnco2 6201	. . . . . . . . 9 ⊢ ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
39		f1ofo 6770	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ–onto→𝐺)
40	39	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ–onto→𝐺)
41		forn 6738	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐺 → ran 𝑓 = 𝐺)
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ran 𝑓 = 𝐺)
43	42	imaeq2d 6009	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
44	38, 43	eqtrid 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
45	44	unieqd 4872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ 𝐺))
46		f1of 6763	. . . . . . . . 9 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ⟶𝐺)
47	12, 15	sstrdi 3947	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
48		fss 6667	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49	46, 47, 48	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
50		fss 6667	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
51	46, 12, 50	syl2anr 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶ran 𝐹)
52		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
53		ffvelcdm 7014	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝐹 ∧ 𝑎 ∈ ℕ) → (𝑓‘𝑎) ∈ ran 𝐹)
54	51, 52, 53	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑎) ∈ ran 𝐹)
55		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
56		ffvelcdm 7014	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝐹 ∧ 𝑏 ∈ ℕ) → (𝑓‘𝑏) ∈ ran 𝐹)
57	51, 55, 56	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑏) ∈ ran 𝐹)
58	1	dyaddisj 25525	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑎) ∈ ran 𝐹 ∧ (𝑓‘𝑏) ∈ ran 𝐹) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) ∨ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
59	54, 57, 58	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) ∨ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
60		fveq2 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑏) → ([,]‘𝑤) = ([,]‘(𝑓‘𝑏)))
61	60	sseq2d 3967	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑏) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏))))
62		eqeq2 2743	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑏) → ((𝑓‘𝑎) = 𝑤 ↔ (𝑓‘𝑎) = (𝑓‘𝑏)))
63	61, 62	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑓‘𝑏) → ((([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤) ↔ (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) → (𝑓‘𝑎) = (𝑓‘𝑏))))
64	46	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶𝐺)
65		ffvelcdm 7014	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝑎 ∈ ℕ) → (𝑓‘𝑎) ∈ 𝐺)
66	64, 52, 65	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑎) ∈ 𝐺)
67		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑓‘𝑎) → ([,]‘𝑧) = ([,]‘(𝑓‘𝑎)))
68	67	sseq1d 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤)))
69		eqeq1 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑎) → (𝑧 = 𝑤 ↔ (𝑓‘𝑎) = 𝑤))
70	68, 69	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
71	70	ralbidv 3155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑓‘𝑎) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
72	71, 2	elrab2 3650	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑎) ∈ 𝐺 ↔ ((𝑓‘𝑎) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
73	72	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑎) ∈ 𝐺 → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤))
74	66, 73	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤))
75		ffvelcdm 7014	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝑏 ∈ ℕ) → (𝑓‘𝑏) ∈ 𝐺)
76	64, 55, 75	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑏) ∈ 𝐺)
77	11, 76	sselid 3932	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑏) ∈ 𝐴)
78	63, 74, 77	rspcdva 3578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) → (𝑓‘𝑎) = (𝑓‘𝑏)))
79		f1of1 6762	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ–1-1→𝐺)
80	79	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ–1-1→𝐺)
81		f1fveq 7196	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1→𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓‘𝑎) = (𝑓‘𝑏) ↔ 𝑎 = 𝑏))
82	80, 81	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓‘𝑎) = (𝑓‘𝑏) ↔ 𝑎 = 𝑏))
83		orc 867	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
84	82, 83	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓‘𝑎) = (𝑓‘𝑏) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
85	78, 84	syld 47	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
86		fveq2 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑎) → ([,]‘𝑤) = ([,]‘(𝑓‘𝑎)))
87	86	sseq2d 3967	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑎) → (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎))))
88		eqeq2 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑎) → ((𝑓‘𝑏) = 𝑤 ↔ (𝑓‘𝑏) = (𝑓‘𝑎)))
89		eqcom 2738	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑏) = (𝑓‘𝑎) ↔ (𝑓‘𝑎) = (𝑓‘𝑏))
90	88, 89	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑎) → ((𝑓‘𝑏) = 𝑤 ↔ (𝑓‘𝑎) = (𝑓‘𝑏)))
91	87, 90	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑓‘𝑎) → ((([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤) ↔ (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) → (𝑓‘𝑎) = (𝑓‘𝑏))))
92		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑓‘𝑏) → ([,]‘𝑧) = ([,]‘(𝑓‘𝑏)))
93	92	sseq1d 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤)))
94		eqeq1 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑏) → (𝑧 = 𝑤 ↔ (𝑓‘𝑏) = 𝑤))
95	93, 94	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
96	95	ralbidv 3155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑓‘𝑏) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
97	96, 2	elrab2 3650	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑏) ∈ 𝐺 ↔ ((𝑓‘𝑏) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
98	97	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑏) ∈ 𝐺 → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤))
99	76, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤))
100	11, 66	sselid 3932	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑎) ∈ 𝐴)
101	91, 99, 100	rspcdva 3578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) → (𝑓‘𝑎) = (𝑓‘𝑏)))
102	101, 84	syld 47	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
103		olc 868	. . . . . . . . . . . . 13 ⊢ ((((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
105	85, 102, 104	3jaod 1431	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) ∨ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
106	59, 105	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
107	106	ralrimivva 3175	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
108		2fveq3 6827	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((,)‘(𝑓‘𝑎)) = ((,)‘(𝑓‘𝑏)))
109	108	disjor 5073	. . . . . . . . 9 ⊢ (Disj 𝑎 ∈ ℕ ((,)‘(𝑓‘𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
110	107, 109	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓‘𝑎)))
111		eqid 2731	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
112	49, 110, 111	uniiccmbl 25519	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ran ([,] ∘ 𝑓) ∈ dom vol)
113	45, 112	eqeltrrd 2832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ([,] “ 𝐺) ∈ dom vol)
114	113	ex 412	. . . . 5 ⊢ (𝜑 → (𝑓:ℕ–1-1-onto→𝐺 → ∪ ([,] “ 𝐺) ∈ dom vol))
115	114	exlimdv 1934	. . . 4 ⊢ (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto→𝐺 → ∪ ([,] “ 𝐺) ∈ dom vol))
116		nnenom 13887	. . . . . 6 ⊢ ℕ ≈ ω
117		ensym 8925	. . . . . 6 ⊢ (𝐺 ≈ ω → ω ≈ 𝐺)
118		entr 8928	. . . . . 6 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
119	116, 117, 118	sylancr 587	. . . . 5 ⊢ (𝐺 ≈ ω → ℕ ≈ 𝐺)
120		bren 8879	. . . . 5 ⊢ (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐺)
121	119, 120	sylib 218	. . . 4 ⊢ (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto→𝐺)
122	115, 121	impel 505	. . 3 ⊢ ((𝜑 ∧ 𝐺 ≈ ω) → ∪ ([,] “ 𝐺) ∈ dom vol)
123		reex 11097	. . . . . . . . 9 ⊢ ℝ ∈ V
124	123, 123	xpex 7686	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
125	124	inex2 5256	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
126	125, 15	ssexi 5260	. . . . . 6 ⊢ ran 𝐹 ∈ V
127		ssdomg 8922	. . . . . 6 ⊢ (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹 → 𝐺 ≼ ran 𝐹))
128	126, 12, 127	mpsyl 68	. . . . 5 ⊢ (𝜑 → 𝐺 ≼ ran 𝐹)
129		omelon 9536	. . . . . . . 8 ⊢ ω ∈ On
130		znnen 16121	. . . . . . . . . . . 12 ⊢ ℤ ≈ ℕ
131	130, 116	entri 8930	. . . . . . . . . . 11 ⊢ ℤ ≈ ω
132		nn0ennn 13886	. . . . . . . . . . . 12 ⊢ ℕ₀ ≈ ℕ
133	132, 116	entri 8930	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ω
134		xpen 9053	. . . . . . . . . . 11 ⊢ ((ℤ ≈ ω ∧ ℕ₀ ≈ ω) → (ℤ × ℕ₀) ≈ (ω × ω))
135	131, 133, 134	mp2an 692	. . . . . . . . . 10 ⊢ (ℤ × ℕ₀) ≈ (ω × ω)
136		xpomen 9906	. . . . . . . . . 10 ⊢ (ω × ω) ≈ ω
137	135, 136	entri 8930	. . . . . . . . 9 ⊢ (ℤ × ℕ₀) ≈ ω
138	137	ensymi 8926	. . . . . . . 8 ⊢ ω ≈ (ℤ × ℕ₀)
139		isnumi 9839	. . . . . . . 8 ⊢ ((ω ∈ On ∧ ω ≈ (ℤ × ℕ₀)) → (ℤ × ℕ₀) ∈ dom card)
140	129, 138, 139	mp2an 692	. . . . . . 7 ⊢ (ℤ × ℕ₀) ∈ dom card
141		ffn 6651	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
142	13, 141	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn (ℤ × ℕ₀)
143		dffn4 6741	. . . . . . . 8 ⊢ (𝐹 Fn (ℤ × ℕ₀) ↔ 𝐹:(ℤ × ℕ₀)–onto→ran 𝐹)
144	142, 143	mpbi 230	. . . . . . 7 ⊢ 𝐹:(ℤ × ℕ₀)–onto→ran 𝐹
145		fodomnum 9948	. . . . . . 7 ⊢ ((ℤ × ℕ₀) ∈ dom card → (𝐹:(ℤ × ℕ₀)–onto→ran 𝐹 → ran 𝐹 ≼ (ℤ × ℕ₀)))
146	140, 144, 145	mp2 9	. . . . . 6 ⊢ ran 𝐹 ≼ (ℤ × ℕ₀)
147		domentr 8935	. . . . . 6 ⊢ ((ran 𝐹 ≼ (ℤ × ℕ₀) ∧ (ℤ × ℕ₀) ≈ ω) → ran 𝐹 ≼ ω)
148	146, 137, 147	mp2an 692	. . . . 5 ⊢ ran 𝐹 ≼ ω
149		domtr 8929	. . . . 5 ⊢ ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
150	128, 148, 149	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐺 ≼ ω)
151		brdom2 8904	. . . 4 ⊢ (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
152	150, 151	sylib 218	. . 3 ⊢ (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
153	37, 122, 152	mpjaodan 960	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ∈ dom vol)
154	4, 153	eqeltrd 2831	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ∀wral 3047 {crab 3395 Vcvv 3436 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 ⟨cop 4582 ∪ cuni 4859 ∪ ciun 4941 Disj wdisj 5058 class class class wbr 5091 × cxp 5614 dom cdm 5616 ran crn 5617 “ cima 5619 ∘ ccom 5620 Oncon0 6306 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 ωcom 7796 1^st c1st 7919 2^nd c2nd 7920 ≈ cen 8866 ≼ cdom 8867 ≺ csdm 8868 Fincfn 8869 cardccrd 9828 ℝcr 11005 1c1 11007 + caddc 11009 ℝ^*cxr 11145 ≤ cle 11147 − cmin 11344 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 (,)cioo 13245 [,]cicc 13248 seqcseq 13908 ↑cexp 13968 abscabs 15141 volcvol 25392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-disj 5059 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-rest 17326 df-topgen 17347 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-top 22810 df-topon 22827 df-bases 22862 df-cmp 23303 df-ovol 25393 df-vol 25394

This theorem is referenced by: opnmbllem 25530

W3C validator