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Theorem dyadmbl 24128
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 24127 . 2 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
5 isfinite 9103 . . . 4 (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6 iccf 12824 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 ffun 6510 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
8 funiunfv 6998 . . . . . 6 (Fun [,] → 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺))
96, 7, 8mp2b 10 . . . . 5 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺)
10 simpr 485 . . . . . 6 ((𝜑𝐺 ∈ Fin) → 𝐺 ∈ Fin)
112ssrab3 4054 . . . . . . . . . . . . . . 15 𝐺𝐴
1211, 3sstrid 3975 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ ran 𝐹)
131dyadf 24119 . . . . . . . . . . . . . . . 16 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
14 frn 6513 . . . . . . . . . . . . . . . 16 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
1513, 14ax-mp 5 . . . . . . . . . . . . . . 15 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
16 inss2 4203 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
1715, 16sstri 3973 . . . . . . . . . . . . . 14 ran 𝐹 ⊆ (ℝ × ℝ)
1812, 17sstrdi 3976 . . . . . . . . . . . . 13 (𝜑𝐺 ⊆ (ℝ × ℝ))
1918adantr 481 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
2019sselda 3964 . . . . . . . . . . 11 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 ∈ (ℝ × ℝ))
21 1st2nd2 7717 . . . . . . . . . . 11 (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2220, 21syl 17 . . . . . . . . . 10 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2322fveq2d 6667 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩))
24 df-ov 7148 . . . . . . . . 9 ((1st𝑛)[,](2nd𝑛)) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩)
2523, 24syl6eqr 2871 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ((1st𝑛)[,](2nd𝑛)))
26 xp1st 7710 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (1st𝑛) ∈ ℝ)
2720, 26syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (1st𝑛) ∈ ℝ)
28 xp2nd 7711 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (2nd𝑛) ∈ ℝ)
2920, 28syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (2nd𝑛) ∈ ℝ)
30 iccmbl 24094 . . . . . . . . 9 (((1st𝑛) ∈ ℝ ∧ (2nd𝑛) ∈ ℝ) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3127, 29, 30syl2anc 584 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3225, 31eqeltrd 2910 . . . . . . 7 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) ∈ dom vol)
3332ralrimiva 3179 . . . . . 6 ((𝜑𝐺 ∈ Fin) → ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
34 finiunmbl 24072 . . . . . 6 ((𝐺 ∈ Fin ∧ ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
3510, 33, 34syl2anc 584 . . . . 5 ((𝜑𝐺 ∈ Fin) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
369, 35eqeltrrid 2915 . . . 4 ((𝜑𝐺 ∈ Fin) → ([,] “ 𝐺) ∈ dom vol)
375, 36sylan2br 594 . . 3 ((𝜑𝐺 ≺ ω) → ([,] “ 𝐺) ∈ dom vol)
38 rnco2 6099 . . . . . . . . 9 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
39 f1ofo 6615 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–onto𝐺)
4039adantl 482 . . . . . . . . . . 11 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–onto𝐺)
41 forn 6586 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐺 → ran 𝑓 = 𝐺)
4240, 41syl 17 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran 𝑓 = 𝐺)
4342imaeq2d 5922 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
4438, 43syl5eq 2865 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
4544unieqd 4840 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
46 f1of 6608 . . . . . . . . 9 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ⟶𝐺)
4712, 15sstrdi 3976 . . . . . . . . 9 (𝜑𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
48 fss 6520 . . . . . . . . 9 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4946, 47, 48syl2anr 596 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
50 fss 6520 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
5146, 12, 50syl2anr 596 . . . . . . . . . . . . 13 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶ran 𝐹)
52 simpl 483 . . . . . . . . . . . . 13 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
53 ffvelrn 6841 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶ran 𝐹𝑎 ∈ ℕ) → (𝑓𝑎) ∈ ran 𝐹)
5451, 52, 53syl2an 595 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ ran 𝐹)
55 simpr 485 . . . . . . . . . . . . 13 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
56 ffvelrn 6841 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶ran 𝐹𝑏 ∈ ℕ) → (𝑓𝑏) ∈ ran 𝐹)
5751, 55, 56syl2an 595 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ ran 𝐹)
581dyaddisj 24124 . . . . . . . . . . . 12 (((𝑓𝑎) ∈ ran 𝐹 ∧ (𝑓𝑏) ∈ ran 𝐹) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
5954, 57, 58syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
60 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → ([,]‘𝑤) = ([,]‘(𝑓𝑏)))
6160sseq2d 3996 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑏) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏))))
62 eqeq2 2830 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑏) → ((𝑓𝑎) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
6361, 62imbi12d 346 . . . . . . . . . . . . . 14 (𝑤 = (𝑓𝑏) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
6446adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶𝐺)
65 ffvelrn 6841 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶𝐺𝑎 ∈ ℕ) → (𝑓𝑎) ∈ 𝐺)
6664, 52, 65syl2an 595 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐺)
67 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → ([,]‘𝑧) = ([,]‘(𝑓𝑎)))
6867sseq1d 3995 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤)))
69 eqeq1 2822 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → (𝑧 = 𝑤 ↔ (𝑓𝑎) = 𝑤))
7068, 69imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7170ralbidv 3194 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑓𝑎) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7271, 2elrab2 3680 . . . . . . . . . . . . . . . 16 ((𝑓𝑎) ∈ 𝐺 ↔ ((𝑓𝑎) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7372simprbi 497 . . . . . . . . . . . . . . 15 ((𝑓𝑎) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
7466, 73syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
75 ffvelrn 6841 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶𝐺𝑏 ∈ ℕ) → (𝑓𝑏) ∈ 𝐺)
7664, 55, 75syl2an 595 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐺)
7711, 76sseldi 3962 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐴)
7863, 74, 77rspcdva 3622 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏)))
79 f1of1 6607 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–1-1𝐺)
8079adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–1-1𝐺)
81 f1fveq 7011 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
8280, 81sylan 580 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
83 orc 861 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
8482, 83syl6bi 254 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
8578, 84syld 47 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
86 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ([,]‘𝑤) = ([,]‘(𝑓𝑎)))
8786sseq2d 3996 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑎) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎))))
88 eqeq2 2830 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑏) = (𝑓𝑎)))
89 eqcom 2825 . . . . . . . . . . . . . . . 16 ((𝑓𝑏) = (𝑓𝑎) ↔ (𝑓𝑎) = (𝑓𝑏))
9088, 89syl6bb 288 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
9187, 90imbi12d 346 . . . . . . . . . . . . . 14 (𝑤 = (𝑓𝑎) → ((([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
92 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → ([,]‘𝑧) = ([,]‘(𝑓𝑏)))
9392sseq1d 3995 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤)))
94 eqeq1 2822 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → (𝑧 = 𝑤 ↔ (𝑓𝑏) = 𝑤))
9593, 94imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9695ralbidv 3194 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑓𝑏) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9796, 2elrab2 3680 . . . . . . . . . . . . . . . 16 ((𝑓𝑏) ∈ 𝐺 ↔ ((𝑓𝑏) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9897simprbi 497 . . . . . . . . . . . . . . 15 ((𝑓𝑏) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
9976, 98syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
10011, 66sseldi 3962 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐴)
10191, 99, 100rspcdva 3622 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏)))
102101, 84syld 47 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
103 olc 862 . . . . . . . . . . . . 13 ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
104103a1i 11 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
10585, 102, 1043jaod 1420 . . . . . . . . . . 11 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
10659, 105mpd 15 . . . . . . . . . 10 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
107106ralrimivva 3188 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
108 2fveq3 6668 . . . . . . . . . 10 (𝑎 = 𝑏 → ((,)‘(𝑓𝑎)) = ((,)‘(𝑓𝑏)))
109108disjor 5037 . . . . . . . . 9 (Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
110107, 109sylibr 235 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)))
111 eqid 2818 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
11249, 110, 111uniiccmbl 24118 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) ∈ dom vol)
11345, 112eqeltrrd 2911 . . . . . 6 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ 𝐺) ∈ dom vol)
114113ex 413 . . . . 5 (𝜑 → (𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
115114exlimdv 1925 . . . 4 (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
116 nnenom 13336 . . . . . 6 ℕ ≈ ω
117 ensym 8546 . . . . . 6 (𝐺 ≈ ω → ω ≈ 𝐺)
118 entr 8549 . . . . . 6 ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
119116, 117, 118sylancr 587 . . . . 5 (𝐺 ≈ ω → ℕ ≈ 𝐺)
120 bren 8506 . . . . 5 (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
121119, 120sylib 219 . . . 4 (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
122115, 121impel 506 . . 3 ((𝜑𝐺 ≈ ω) → ([,] “ 𝐺) ∈ dom vol)
123 reex 10616 . . . . . . . . 9 ℝ ∈ V
124123, 123xpex 7465 . . . . . . . 8 (ℝ × ℝ) ∈ V
125124inex2 5213 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ∈ V
126125, 15ssexi 5217 . . . . . 6 ran 𝐹 ∈ V
127 ssdomg 8543 . . . . . 6 (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹𝐺 ≼ ran 𝐹))
128126, 12, 127mpsyl 68 . . . . 5 (𝜑𝐺 ≼ ran 𝐹)
129 omelon 9097 . . . . . . . 8 ω ∈ On
130 znnen 15553 . . . . . . . . . . . 12 ℤ ≈ ℕ
131130, 116entri 8551 . . . . . . . . . . 11 ℤ ≈ ω
132 nn0ennn 13335 . . . . . . . . . . . 12 0 ≈ ℕ
133132, 116entri 8551 . . . . . . . . . . 11 0 ≈ ω
134 xpen 8668 . . . . . . . . . . 11 ((ℤ ≈ ω ∧ ℕ0 ≈ ω) → (ℤ × ℕ0) ≈ (ω × ω))
135131, 133, 134mp2an 688 . . . . . . . . . 10 (ℤ × ℕ0) ≈ (ω × ω)
136 xpomen 9429 . . . . . . . . . 10 (ω × ω) ≈ ω
137135, 136entri 8551 . . . . . . . . 9 (ℤ × ℕ0) ≈ ω
138137ensymi 8547 . . . . . . . 8 ω ≈ (ℤ × ℕ0)
139 isnumi 9363 . . . . . . . 8 ((ω ∈ On ∧ ω ≈ (ℤ × ℕ0)) → (ℤ × ℕ0) ∈ dom card)
140129, 138, 139mp2an 688 . . . . . . 7 (ℤ × ℕ0) ∈ dom card
141 ffn 6507 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14213, 141ax-mp 5 . . . . . . . 8 𝐹 Fn (ℤ × ℕ0)
143 dffn4 6589 . . . . . . . 8 (𝐹 Fn (ℤ × ℕ0) ↔ 𝐹:(ℤ × ℕ0)–onto→ran 𝐹)
144142, 143mpbi 231 . . . . . . 7 𝐹:(ℤ × ℕ0)–onto→ran 𝐹
145 fodomnum 9471 . . . . . . 7 ((ℤ × ℕ0) ∈ dom card → (𝐹:(ℤ × ℕ0)–onto→ran 𝐹 → ran 𝐹 ≼ (ℤ × ℕ0)))
146140, 144, 145mp2 9 . . . . . 6 ran 𝐹 ≼ (ℤ × ℕ0)
147 domentr 8556 . . . . . 6 ((ran 𝐹 ≼ (ℤ × ℕ0) ∧ (ℤ × ℕ0) ≈ ω) → ran 𝐹 ≼ ω)
148146, 137, 147mp2an 688 . . . . 5 ran 𝐹 ≼ ω
149 domtr 8550 . . . . 5 ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
150128, 148, 149sylancl 586 . . . 4 (𝜑𝐺 ≼ ω)
151 brdom2 8527 . . . 4 (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
152150, 151sylib 219 . . 3 (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
15337, 122, 152mpjaodan 952 . 2 (𝜑 ([,] “ 𝐺) ∈ dom vol)
1544, 153eqeltrd 2910 1 (𝜑 ([,] “ 𝐴) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3o 1078   = wceq 1528  wex 1771  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  cop 4563   cuni 4830   ciun 4910  Disj wdisj 5022   class class class wbr 5057   × cxp 5546  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  Oncon0 6184  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  ωcom 7569  1st c1st 7676  2nd c2nd 7677  cen 8494  cdom 8495  csdm 8496  Fincfn 8497  cardccrd 9352  cr 10524  1c1 10526   + caddc 10528  *cxr 10662  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  (,)cioo 12726  [,]cicc 12729  seqcseq 13357  cexp 13417  abscabs 14581  volcvol 23991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992  df-vol 23993
This theorem is referenced by:  opnmbllem  24129
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