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Theorem dyadmbl 25901

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 25900	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))
5		isfinite 9637	. . . 4 ⊢ (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6		iccf 13560	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
7		ffun 6704	. . . . . 6 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
8		funiunfv 7244	. . . . . 6 ⊢ (Fun [,] → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) = ∪ ([,] “ 𝐺))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) = ∪ ([,] “ 𝐺)
10		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → 𝐺 ∈ Fin)
11	2	ssrab3 4030	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ⊆ 𝐴
12	11, 3	sstrid 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ ran 𝐹)
13	1	dyadf 25892	. . . . . . . . . . . . . . . 16 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
14		frn 6709	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
15	13, 14	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
16		inss2 4183	. . . . . . . . . . . . . . 15 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
17	15, 16	sstri 3940	. . . . . . . . . . . . . 14 ⊢ ran 𝐹 ⊆ (ℝ × ℝ)
18	12, 17	sstrdi 3943	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ (ℝ × ℝ))
19	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
20	19	sselda 3931	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → 𝑛 ∈ (ℝ × ℝ))
21		1st2nd2 8029	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
22	20, 21	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
23	22	fveq2d 6881	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) = ([,]‘⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩))
24		df-ov 7415	. . . . . . . . 9 ⊢ ((1^st ‘𝑛)[,](2^nd ‘𝑛)) = ([,]‘⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
25	23, 24	eqtr4di 2814	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) = ((1^st ‘𝑛)[,](2^nd ‘𝑛)))
26		xp1st 8022	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℝ × ℝ) → (1^st ‘𝑛) ∈ ℝ)
27	20, 26	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → (1^st ‘𝑛) ∈ ℝ)
28		xp2nd 8023	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℝ × ℝ) → (2^nd ‘𝑛) ∈ ℝ)
29	20, 28	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → (2^nd ‘𝑛) ∈ ℝ)
30		iccmbl 25867	. . . . . . . . 9 ⊢ (((1^st ‘𝑛) ∈ ℝ ∧ (2^nd ‘𝑛) ∈ ℝ) → ((1^st ‘𝑛)[,](2^nd ‘𝑛)) ∈ dom vol)
31	27, 29, 30	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ((1^st ‘𝑛)[,](2^nd ‘𝑛)) ∈ dom vol)
32	25, 31	eqeltrd 2861	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 ∈ Fin) ∧ 𝑛 ∈ 𝐺) → ([,]‘𝑛) ∈ dom vol)
33	32	ralrimiva 3155	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∀𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
34		finiunmbl 25845	. . . . . 6 ⊢ ((𝐺 ∈ Fin ∧ ∀𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol) → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
35	10, 33, 34	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∪ 𝑛 ∈ 𝐺 ([,]‘𝑛) ∈ dom vol)
36	9, 35	eqeltrrid 2866	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ Fin) → ∪ ([,] “ 𝐺) ∈ dom vol)
37	5, 36	sylan2br 607	. . 3 ⊢ ((𝜑 ∧ 𝐺 ≺ ω) → ∪ ([,] “ 𝐺) ∈ dom vol)
38		rnco2 6248	. . . . . . . . 9 ⊢ ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
39		f1ofo 6824	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ–onto→𝐺)
40	39	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ–onto→𝐺)
41		forn 6791	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐺 → ran 𝑓 = 𝐺)
42	40, 41	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ran 𝑓 = 𝐺)
43	42	imaeq2d 6054	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
44	38, 43	eqtrid 2808	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
45	44	unieqd 4880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ 𝐺))
46		f1of 6816	. . . . . . . . 9 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ⟶𝐺)
47	12, 15	sstrdi 3943	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
48		fss 6718	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49	46, 47, 48	syl2anr 609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
50		fss 6718	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
51	46, 12, 50	syl2anr 609	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶ran 𝐹)
52		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
53		ffvelcdm 7073	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝐹 ∧ 𝑎 ∈ ℕ) → (𝑓‘𝑎) ∈ ran 𝐹)
54	51, 52, 53	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑎) ∈ ran 𝐹)
55		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
56		ffvelcdm 7073	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝐹 ∧ 𝑏 ∈ ℕ) → (𝑓‘𝑏) ∈ ran 𝐹)
57	51, 55, 56	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑏) ∈ ran 𝐹)
58	1	dyaddisj 25897	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑏) → ([,]‘𝑤) = ([,]‘(𝑓‘𝑏)))
61	60	sseq2d 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑏) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏))))
62		eqeq2 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑏) → ((𝑓‘𝑎) = 𝑤 ↔ (𝑓‘𝑎) = (𝑓‘𝑏)))
63	61, 62	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑓‘𝑏) → ((([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤) ↔ (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) → (𝑓‘𝑎) = (𝑓‘𝑏))))
64	46	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ⟶𝐺)
65		ffvelcdm 7073	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝑎 ∈ ℕ) → (𝑓‘𝑎) ∈ 𝐺)
66	64, 52, 65	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑎) ∈ 𝐺)
67		fveq2 6877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑓‘𝑎) → ([,]‘𝑧) = ([,]‘(𝑓‘𝑎)))
68	67	sseq1d 3962	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤)))
69		eqeq1 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑎) → (𝑧 = 𝑤 ↔ (𝑓‘𝑎) = 𝑤))
70	68, 69	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
71	70	ralbidv 3186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑓‘𝑎) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
72	71, 2	elrab2 3649	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑎) ∈ 𝐺 ↔ ((𝑓‘𝑎) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤)))
73	72	simprbi 503	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑎) ∈ 𝐺 → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤))
74	66, 73	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘𝑤) → (𝑓‘𝑎) = 𝑤))
75		ffvelcdm 7073	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶𝐺 ∧ 𝑏 ∈ ℕ) → (𝑓‘𝑏) ∈ 𝐺)
76	64, 55, 75	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓‘𝑏) ∈ 𝐺)
77	11, 76	sselid 3929	. . . . . . . . . . . . . 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 6815	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐺 → 𝑓:ℕ–1-1→𝐺)
80	79	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → 𝑓:ℕ–1-1→𝐺)
81		f1fveq 7258	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1→𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓‘𝑎) = (𝑓‘𝑏) ↔ 𝑎 = 𝑏))
82	80, 81	sylan 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓‘𝑎) = (𝑓‘𝑏) ↔ 𝑎 = 𝑏))
83		orc 881	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
84	82, 83	biimtrdi 256	. . . . . . . . . . . . 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 48	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑎)) ⊆ ([,]‘(𝑓‘𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
86		fveq2 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑎) → ([,]‘𝑤) = ([,]‘(𝑓‘𝑎)))
87	86	sseq2d 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑎) → (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎))))
88		eqeq2 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓‘𝑎) → ((𝑓‘𝑏) = 𝑤 ↔ (𝑓‘𝑏) = (𝑓‘𝑎)))
89		eqcom 2768	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑏) = (𝑓‘𝑎) ↔ (𝑓‘𝑎) = (𝑓‘𝑏))
90	88, 89	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑓‘𝑎) → ((𝑓‘𝑏) = 𝑤 ↔ (𝑓‘𝑎) = (𝑓‘𝑏)))
91	87, 90	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑓‘𝑎) → ((([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤) ↔ (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) → (𝑓‘𝑎) = (𝑓‘𝑏))))
92		fveq2 6877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑓‘𝑏) → ([,]‘𝑧) = ([,]‘(𝑓‘𝑏)))
93	92	sseq1d 3962	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤)))
94		eqeq1 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑓‘𝑏) → (𝑧 = 𝑤 ↔ (𝑓‘𝑏) = 𝑤))
95	93, 94	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑓‘𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
96	95	ralbidv 3186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑓‘𝑏) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
97	96, 2	elrab2 3649	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑏) ∈ 𝐺 ↔ ((𝑓‘𝑏) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤)))
98	97	simprbi 503	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑏) ∈ 𝐺 → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤))
99	76, 98	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘𝑤) → (𝑓‘𝑏) = 𝑤))
100	11, 66	sselid 3929	. . . . . . . . . . . . . 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 48	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓‘𝑏)) ⊆ ([,]‘(𝑓‘𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅)))
103		olc 882	. . . . . . . . . . . . 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 1456	. . . . . . . . . . 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 16	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
107	106	ralrimivva 3206	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
108		2fveq3 6882	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((,)‘(𝑓‘𝑎)) = ((,)‘(𝑓‘𝑏)))
109	108	disjor 5085	. . . . . . . . 9 ⊢ (Disj 𝑎 ∈ ℕ ((,)‘(𝑓‘𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓‘𝑎)) ∩ ((,)‘(𝑓‘𝑏))) = ∅))
110	107, 109	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓‘𝑎)))
111		eqid 2761	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
112	49, 110, 111	uniiccmbl 25891	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ran ([,] ∘ 𝑓) ∈ dom vol)
113	45, 112	eqeltrrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→𝐺) → ∪ ([,] “ 𝐺) ∈ dom vol)
114	113	ex 418	. . . . 5 ⊢ (𝜑 → (𝑓:ℕ–1-1-onto→𝐺 → ∪ ([,] “ 𝐺) ∈ dom vol))
115	114	exlimdv 1966	. . . 4 ⊢ (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto→𝐺 → ∪ ([,] “ 𝐺) ∈ dom vol))
116		nnenom 14103	. . . . . 6 ⊢ ℕ ≈ ω
117		ensym 9014	. . . . . 6 ⊢ (𝐺 ≈ ω → ω ≈ 𝐺)
118		entr 9017	. . . . . 6 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
119	116, 117, 118	sylancr 599	. . . . 5 ⊢ (𝐺 ≈ ω → ℕ ≈ 𝐺)
120		bren 8967	. . . . 5 ⊢ (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐺)
121	119, 120	sylib 221	. . . 4 ⊢ (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto→𝐺)
122	115, 121	impel 515	. . 3 ⊢ ((𝜑 ∧ 𝐺 ≈ ω) → ∪ ([,] “ 𝐺) ∈ dom vol)
123		reex 11272	. . . . . . . . 9 ⊢ ℝ ∈ V
124	123, 123	xpex 7756	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
125	124	inex2 5278	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
126	125, 15	ssexi 5284	. . . . . 6 ⊢ ran 𝐹 ∈ V
127		ssdomg 9011	. . . . . 6 ⊢ (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹 → 𝐺 ≼ ran 𝐹))
128	126, 12, 127	mpsyl 69	. . . . 5 ⊢ (𝜑 → 𝐺 ≼ ran 𝐹)
129		omelon 9631	. . . . . . . 8 ⊢ ω ∈ On
130		znnen 16360	. . . . . . . . . . . 12 ⊢ ℤ ≈ ℕ
131	130, 116	entri 9019	. . . . . . . . . . 11 ⊢ ℤ ≈ ω
132		nn0ennn 14102	. . . . . . . . . . . 12 ⊢ ℕ₀ ≈ ℕ
133	132, 116	entri 9019	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ω
134		xpen 9143	. . . . . . . . . . 11 ⊢ ((ℤ ≈ ω ∧ ℕ₀ ≈ ω) → (ℤ × ℕ₀) ≈ (ω × ω))
135	131, 133, 134	mp2an 705	. . . . . . . . . 10 ⊢ (ℤ × ℕ₀) ≈ (ω × ω)
136		xpomen 10075	. . . . . . . . . 10 ⊢ (ω × ω) ≈ ω
137	135, 136	entri 9019	. . . . . . . . 9 ⊢ (ℤ × ℕ₀) ≈ ω
138	137	ensymi 9015	. . . . . . . 8 ⊢ ω ≈ (ℤ × ℕ₀)
139		isnumi 10008	. . . . . . . 8 ⊢ ((ω ∈ On ∧ ω ≈ (ℤ × ℕ₀)) → (ℤ × ℕ₀) ∈ dom card)
140	129, 138, 139	mp2an 705	. . . . . . 7 ⊢ (ℤ × ℕ₀) ∈ dom card
141		ffn 6701	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
142	13, 141	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn (ℤ × ℕ₀)
143		dffn4 6794	. . . . . . . 8 ⊢ (𝐹 Fn (ℤ × ℕ₀) ↔ 𝐹:(ℤ × ℕ₀)–onto→ran 𝐹)
144	142, 143	mpbi 233	. . . . . . 7 ⊢ 𝐹:(ℤ × ℕ₀)–onto→ran 𝐹
145		fodomnum 10117	. . . . . . 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 9024	. . . . . 6 ⊢ ((ran 𝐹 ≼ (ℤ × ℕ₀) ∧ (ℤ × ℕ₀) ≈ ω) → ran 𝐹 ≼ ω)
148	146, 137, 147	mp2an 705	. . . . 5 ⊢ ran 𝐹 ≼ ω
149		domtr 9018	. . . . 5 ⊢ ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
150	128, 148, 149	sylancl 598	. . . 4 ⊢ (𝜑 → 𝐺 ≼ ω)
151		brdom2 8993	. . . 4 ⊢ (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
152	150, 151	sylib 221	. . 3 ⊢ (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
153	37, 122, 152	mpjaodan 973	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ∈ dom vol)
154	4, 153	eqeltrd 2861	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ∈ dom vol)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∀wral 3077 {crab 3413 Vcvv 3451 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 ⟨cop 4590 ∪ cuni 4867 ∪ ciun 4951 Disj wdisj 5070 class class class wbr 5103 × cxp 5649 dom cdm 5651 ran crn 5652 “ cima 5654 ∘ ccom 5655 Oncon0 6355 Fun wfun 6525 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 –onto→wfo 6529 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 ωcom 7866 1^st c1st 7988 2^nd c2nd 7989 ≈ cen 8954 ≼ cdom 8955 ≺ csdm 8956 Fincfn 8957 cardccrd 9997 ℝcr 11180 1c1 11182 + caddc 11184 ℝ^*cxr 11323 ≤ cle 11325 − cmin 11522 / cdiv 11954 ℕcn 12316 2c2 12378 ℕ₀cn0 12587 ℤcz 12674 (,)cioo 13457 [,]cicc 13460 seqcseq 14124 ↑cexp 14184 abscabs 15381 volcvol 25764

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-omul 8465 df-er 8701 df-map 8833 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-acn 10004 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-n0 12588 df-z 12675 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-clim 15635 df-rlim 15636 df-sum 15834 df-rest 17573 df-topgen 17594 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-top 23192 df-topon 23209 df-bases 23244 df-cmp 23685 df-ovol 25765 df-vol 25766

This theorem is used by: opnmbllem 25902

W3C validator