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Theorem dyadmbl 23581
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 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
dyadmbl.3 (𝜑𝐴 ⊆ 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.
StepHypRef Expression
1 dyadmbl.1 . . 3 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
2 dyadmbl.2 . . 3 𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
3 dyadmbl.3 . . 3 (𝜑𝐴 ⊆ ran 𝐹)
41, 2, 3dyadmbllem 23580 . 2 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
5 isfinite 8711 . . . 4 (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6 iccf 12471 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 ffun 6186 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
8 funiunfv 6647 . . . . . 6 (Fun [,] → 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺))
96, 7, 8mp2b 10 . . . . 5 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺)
10 simpr 471 . . . . . 6 ((𝜑𝐺 ∈ Fin) → 𝐺 ∈ Fin)
11 ssrab2 3836 . . . . . . . . . . . . . . . 16 {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} ⊆ 𝐴
122, 11eqsstri 3784 . . . . . . . . . . . . . . 15 𝐺𝐴
1312, 3syl5ss 3763 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ ran 𝐹)
141dyadf 23572 . . . . . . . . . . . . . . . 16 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
15 frn 6191 . . . . . . . . . . . . . . . 16 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
17 inss2 3982 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
1816, 17sstri 3761 . . . . . . . . . . . . . 14 ran 𝐹 ⊆ (ℝ × ℝ)
1913, 18syl6ss 3764 . . . . . . . . . . . . 13 (𝜑𝐺 ⊆ (ℝ × ℝ))
2019adantr 466 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
2120sselda 3752 . . . . . . . . . . 11 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 ∈ (ℝ × ℝ))
22 1st2nd2 7352 . . . . . . . . . . 11 (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2321, 22syl 17 . . . . . . . . . 10 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2423fveq2d 6334 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩))
25 df-ov 6794 . . . . . . . . 9 ((1st𝑛)[,](2nd𝑛)) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩)
2624, 25syl6eqr 2823 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ((1st𝑛)[,](2nd𝑛)))
27 xp1st 7345 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (1st𝑛) ∈ ℝ)
2821, 27syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (1st𝑛) ∈ ℝ)
29 xp2nd 7346 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (2nd𝑛) ∈ ℝ)
3021, 29syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (2nd𝑛) ∈ ℝ)
31 iccmbl 23547 . . . . . . . . 9 (((1st𝑛) ∈ ℝ ∧ (2nd𝑛) ∈ ℝ) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3228, 30, 31syl2anc 573 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3326, 32eqeltrd 2850 . . . . . . 7 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) ∈ dom vol)
3433ralrimiva 3115 . . . . . 6 ((𝜑𝐺 ∈ Fin) → ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
35 finiunmbl 23525 . . . . . 6 ((𝐺 ∈ Fin ∧ ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
3610, 34, 35syl2anc 573 . . . . 5 ((𝜑𝐺 ∈ Fin) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
379, 36syl5eqelr 2855 . . . 4 ((𝜑𝐺 ∈ Fin) → ([,] “ 𝐺) ∈ dom vol)
385, 37sylan2br 582 . . 3 ((𝜑𝐺 ≺ ω) → ([,] “ 𝐺) ∈ dom vol)
39 nnenom 12980 . . . . . . 7 ℕ ≈ ω
40 ensym 8156 . . . . . . 7 (𝐺 ≈ ω → ω ≈ 𝐺)
41 entr 8159 . . . . . . 7 ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
4239, 40, 41sylancr 575 . . . . . 6 (𝐺 ≈ ω → ℕ ≈ 𝐺)
43 bren 8116 . . . . . 6 (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
4442, 43sylib 208 . . . . 5 (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
45 rnco2 5784 . . . . . . . . . 10 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
46 f1ofo 6283 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–onto𝐺)
4746adantl 467 . . . . . . . . . . . 12 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–onto𝐺)
48 forn 6257 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐺 → ran 𝑓 = 𝐺)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran 𝑓 = 𝐺)
5049imaeq2d 5605 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
5145, 50syl5eq 2817 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
5251unieqd 4584 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
53 f1of 6276 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ⟶𝐺)
5413, 16syl6ss 3764 . . . . . . . . . 10 (𝜑𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
55 fss 6194 . . . . . . . . . 10 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5653, 54, 55syl2anr 584 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
57 fss 6194 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
5853, 13, 57syl2anr 584 . . . . . . . . . . . . . 14 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶ran 𝐹)
59 simpl 468 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
60 ffvelrn 6498 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑎 ∈ ℕ) → (𝑓𝑎) ∈ ran 𝐹)
6158, 59, 60syl2an 583 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ ran 𝐹)
62 simpr 471 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
63 ffvelrn 6498 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑏 ∈ ℕ) → (𝑓𝑏) ∈ ran 𝐹)
6458, 62, 63syl2an 583 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ ran 𝐹)
651dyaddisj 23577 . . . . . . . . . . . . 13 (((𝑓𝑎) ∈ ran 𝐹 ∧ (𝑓𝑏) ∈ ran 𝐹) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
6661, 64, 65syl2anc 573 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
67 fveq2 6330 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑏) → ([,]‘𝑤) = ([,]‘(𝑓𝑏)))
6867sseq2d 3782 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏))))
69 eqeq2 2782 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → ((𝑓𝑎) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
7068, 69imbi12d 333 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑏) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
7153adantl 467 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶𝐺)
72 ffvelrn 6498 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑎 ∈ ℕ) → (𝑓𝑎) ∈ 𝐺)
7371, 59, 72syl2an 583 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐺)
74 fveq2 6330 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑎) → ([,]‘𝑧) = ([,]‘(𝑓𝑎)))
7574sseq1d 3781 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤)))
76 eqeq1 2775 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (𝑧 = 𝑤 ↔ (𝑓𝑎) = 𝑤))
7775, 76imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7877ralbidv 3135 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑎) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7978, 2elrab2 3518 . . . . . . . . . . . . . . . . 17 ((𝑓𝑎) ∈ 𝐺 ↔ ((𝑓𝑎) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
8079simprbi 484 . . . . . . . . . . . . . . . 16 ((𝑓𝑎) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
8173, 80syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
82 ffvelrn 6498 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑏 ∈ ℕ) → (𝑓𝑏) ∈ 𝐺)
8371, 62, 82syl2an 583 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐺)
8412, 83sseldi 3750 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐴)
8570, 81, 84rspcdva 3466 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏)))
86 f1of1 6275 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–1-1𝐺)
8786adantl 467 . . . . . . . . . . . . . . . 16 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–1-1𝐺)
88 f1fveq 6660 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
8987, 88sylan 569 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
90 orc 856 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
9189, 90syl6bi 243 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
9285, 91syld 47 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
93 fveq2 6330 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → ([,]‘𝑤) = ([,]‘(𝑓𝑎)))
9493sseq2d 3782 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎))))
95 eqeq2 2782 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑏) = (𝑓𝑎)))
96 eqcom 2778 . . . . . . . . . . . . . . . . 17 ((𝑓𝑏) = (𝑓𝑎) ↔ (𝑓𝑎) = (𝑓𝑏))
9795, 96syl6bb 276 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
9894, 97imbi12d 333 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑎) → ((([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
99 fveq2 6330 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑏) → ([,]‘𝑧) = ([,]‘(𝑓𝑏)))
10099sseq1d 3781 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤)))
101 eqeq1 2775 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (𝑧 = 𝑤 ↔ (𝑓𝑏) = 𝑤))
102100, 101imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
103102ralbidv 3135 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑏) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
104103, 2elrab2 3518 . . . . . . . . . . . . . . . . 17 ((𝑓𝑏) ∈ 𝐺 ↔ ((𝑓𝑏) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
105104simprbi 484 . . . . . . . . . . . . . . . 16 ((𝑓𝑏) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
10683, 105syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
10712, 73sseldi 3750 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐴)
10898, 106, 107rspcdva 3466 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏)))
109108, 91syld 47 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
110 olc 857 . . . . . . . . . . . . . 14 ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
111110a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
11292, 109, 1113jaod 1540 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
11366, 112mpd 15 . . . . . . . . . . 11 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
114113ralrimivva 3120 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
115 fveq2 6330 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑓𝑎) = (𝑓𝑏))
116115fveq2d 6334 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((,)‘(𝑓𝑎)) = ((,)‘(𝑓𝑏)))
117116disjor 4768 . . . . . . . . . 10 (Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
118114, 117sylibr 224 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)))
119 eqid 2771 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
12056, 118, 119uniiccmbl 23571 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) ∈ dom vol)
12152, 120eqeltrrd 2851 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ 𝐺) ∈ dom vol)
122121ex 397 . . . . . 6 (𝜑 → (𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
123122exlimdv 2013 . . . . 5 (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
12444, 123syl5 34 . . . 4 (𝜑 → (𝐺 ≈ ω → ([,] “ 𝐺) ∈ dom vol))
125124imp 393 . . 3 ((𝜑𝐺 ≈ ω) → ([,] “ 𝐺) ∈ dom vol)
126 reex 10227 . . . . . . . . 9 ℝ ∈ V
127126, 126xpex 7107 . . . . . . . 8 (ℝ × ℝ) ∈ V
128127inex2 4934 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ∈ V
129128, 16ssexi 4937 . . . . . 6 ran 𝐹 ∈ V
130 ssdomg 8153 . . . . . 6 (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹𝐺 ≼ ran 𝐹))
131129, 13, 130mpsyl 68 . . . . 5 (𝜑𝐺 ≼ ran 𝐹)
132 omelon 8705 . . . . . . . 8 ω ∈ On
133 znnen 15140 . . . . . . . . . . . 12 ℤ ≈ ℕ
134133, 39entri 8161 . . . . . . . . . . 11 ℤ ≈ ω
135 nn0ennn 12979 . . . . . . . . . . . 12 0 ≈ ℕ
136135, 39entri 8161 . . . . . . . . . . 11 0 ≈ ω
137 xpen 8277 . . . . . . . . . . 11 ((ℤ ≈ ω ∧ ℕ0 ≈ ω) → (ℤ × ℕ0) ≈ (ω × ω))
138134, 136, 137mp2an 672 . . . . . . . . . 10 (ℤ × ℕ0) ≈ (ω × ω)
139 xpomen 9036 . . . . . . . . . 10 (ω × ω) ≈ ω
140138, 139entri 8161 . . . . . . . . 9 (ℤ × ℕ0) ≈ ω
141140ensymi 8157 . . . . . . . 8 ω ≈ (ℤ × ℕ0)
142 isnumi 8970 . . . . . . . 8 ((ω ∈ On ∧ ω ≈ (ℤ × ℕ0)) → (ℤ × ℕ0) ∈ dom card)
143132, 141, 142mp2an 672 . . . . . . 7 (ℤ × ℕ0) ∈ dom card
144 ffn 6183 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14514, 144ax-mp 5 . . . . . . . 8 𝐹 Fn (ℤ × ℕ0)
146 dffn4 6260 . . . . . . . 8 (𝐹 Fn (ℤ × ℕ0) ↔ 𝐹:(ℤ × ℕ0)–onto→ran 𝐹)
147145, 146mpbi 220 . . . . . . 7 𝐹:(ℤ × ℕ0)–onto→ran 𝐹
148 fodomnum 9078 . . . . . . 7 ((ℤ × ℕ0) ∈ dom card → (𝐹:(ℤ × ℕ0)–onto→ran 𝐹 → ran 𝐹 ≼ (ℤ × ℕ0)))
149143, 147, 148mp2 9 . . . . . 6 ran 𝐹 ≼ (ℤ × ℕ0)
150 domentr 8166 . . . . . 6 ((ran 𝐹 ≼ (ℤ × ℕ0) ∧ (ℤ × ℕ0) ≈ ω) → ran 𝐹 ≼ ω)
151149, 140, 150mp2an 672 . . . . 5 ran 𝐹 ≼ ω
152 domtr 8160 . . . . 5 ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
153131, 151, 152sylancl 574 . . . 4 (𝜑𝐺 ≼ ω)
154 brdom2 8137 . . . 4 (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
155153, 154sylib 208 . . 3 (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
15638, 125, 155mpjaodan 943 . 2 (𝜑 ([,] “ 𝐺) ∈ dom vol)
1574, 156eqeltrd 2850 1 (𝜑 ([,] “ 𝐴) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3o 1070   = wceq 1631  wex 1852  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  cop 4322   cuni 4574   ciun 4654  Disj wdisj 4754   class class class wbr 4786   × cxp 5247  dom cdm 5249  ran crn 5250  cima 5252  ccom 5253  Oncon0 5864  Fun wfun 6023   Fn wfn 6024  wf 6025  1-1wf1 6026  ontowfo 6027  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  cmpt2 6793  ωcom 7210  1st c1st 7311  2nd c2nd 7312  cen 8104  cdom 8105  csdm 8106  Fincfn 8107  cardccrd 8959  cr 10135  1c1 10137   + caddc 10139  *cxr 10273  cle 10275  cmin 10466   / cdiv 10884  cn 11220  2c2 11270  0cn0 11492  cz 11577  (,)cioo 12373  [,]cicc 12376  seqcseq 13001  cexp 13060  abscabs 14175  volcvol 23444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-omul 7716  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-fi 8471  df-sup 8502  df-inf 8503  df-oi 8569  df-card 8963  df-acn 8966  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-n0 11493  df-z 11578  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12144  df-xadd 12145  df-xmul 12146  df-ioo 12377  df-ico 12379  df-icc 12380  df-fz 12527  df-fzo 12667  df-fl 12794  df-seq 13002  df-exp 13061  df-hash 13315  df-cj 14040  df-re 14041  df-im 14042  df-sqrt 14176  df-abs 14177  df-clim 14420  df-rlim 14421  df-sum 14618  df-rest 16284  df-topgen 16305  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-top 20912  df-topon 20929  df-bases 20964  df-cmp 21404  df-ovol 23445  df-vol 23446
This theorem is referenced by:  opnmbllem  23582
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