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Theorem dyadmbl 24135
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 24134 . 2 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
5 isfinite 9109 . . . 4 (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6 iccf 12831 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 ffun 6516 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
8 funiunfv 7003 . . . . . 6 (Fun [,] → 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺))
96, 7, 8mp2b 10 . . . . 5 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺)
10 simpr 485 . . . . . 6 ((𝜑𝐺 ∈ Fin) → 𝐺 ∈ Fin)
112ssrab3 4061 . . . . . . . . . . . . . . 15 𝐺𝐴
1211, 3sstrid 3982 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ ran 𝐹)
131dyadf 24126 . . . . . . . . . . . . . . . 16 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
14 frn 6519 . . . . . . . . . . . . . . . 16 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
1513, 14ax-mp 5 . . . . . . . . . . . . . . 15 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
16 inss2 4210 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
1715, 16sstri 3980 . . . . . . . . . . . . . 14 ran 𝐹 ⊆ (ℝ × ℝ)
1812, 17sstrdi 3983 . . . . . . . . . . . . 13 (𝜑𝐺 ⊆ (ℝ × ℝ))
1918adantr 481 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
2019sselda 3971 . . . . . . . . . . 11 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 ∈ (ℝ × ℝ))
21 1st2nd2 7724 . . . . . . . . . . 11 (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2220, 21syl 17 . . . . . . . . . 10 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2322fveq2d 6673 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩))
24 df-ov 7153 . . . . . . . . 9 ((1st𝑛)[,](2nd𝑛)) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩)
2523, 24syl6eqr 2879 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ((1st𝑛)[,](2nd𝑛)))
26 xp1st 7717 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (1st𝑛) ∈ ℝ)
2720, 26syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (1st𝑛) ∈ ℝ)
28 xp2nd 7718 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (2nd𝑛) ∈ ℝ)
2920, 28syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (2nd𝑛) ∈ ℝ)
30 iccmbl 24101 . . . . . . . . 9 (((1st𝑛) ∈ ℝ ∧ (2nd𝑛) ∈ ℝ) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3127, 29, 30syl2anc 584 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3225, 31eqeltrd 2918 . . . . . . 7 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) ∈ dom vol)
3332ralrimiva 3187 . . . . . 6 ((𝜑𝐺 ∈ Fin) → ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
34 finiunmbl 24079 . . . . . 6 ((𝐺 ∈ Fin ∧ ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
3510, 33, 34syl2anc 584 . . . . 5 ((𝜑𝐺 ∈ Fin) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
369, 35eqeltrrid 2923 . . . 4 ((𝜑𝐺 ∈ Fin) → ([,] “ 𝐺) ∈ dom vol)
375, 36sylan2br 594 . . 3 ((𝜑𝐺 ≺ ω) → ([,] “ 𝐺) ∈ dom vol)
38 rnco2 6105 . . . . . . . . 9 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
39 f1ofo 6621 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–onto𝐺)
4039adantl 482 . . . . . . . . . . 11 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–onto𝐺)
41 forn 6592 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐺 → ran 𝑓 = 𝐺)
4240, 41syl 17 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran 𝑓 = 𝐺)
4342imaeq2d 5928 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
4438, 43syl5eq 2873 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
4544unieqd 4847 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
46 f1of 6614 . . . . . . . . 9 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ⟶𝐺)
4712, 15sstrdi 3983 . . . . . . . . 9 (𝜑𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
48 fss 6526 . . . . . . . . 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 6526 . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . 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 24131 . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → ([,]‘𝑤) = ([,]‘(𝑓𝑏)))
6160sseq2d 4003 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑏) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏))))
62 eqeq2 2838 . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → ([,]‘𝑧) = ([,]‘(𝑓𝑎)))
6867sseq1d 4002 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤)))
69 eqeq1 2830 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → (𝑧 = 𝑤 ↔ (𝑓𝑎) = 𝑤))
7068, 69imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7170ralbidv 3202 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑓𝑎) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7271, 2elrab2 3687 . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐴)
7863, 74, 77rspcdva 3629 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏)))
79 f1of1 6613 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–1-1𝐺)
8079adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–1-1𝐺)
81 f1fveq 7016 . . . . . . . . . . . . . . 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 863 . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ([,]‘𝑤) = ([,]‘(𝑓𝑎)))
8786sseq2d 4003 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑎) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎))))
88 eqeq2 2838 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑏) = (𝑓𝑎)))
89 eqcom 2833 . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → ([,]‘𝑧) = ([,]‘(𝑓𝑏)))
9392sseq1d 4002 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤)))
94 eqeq1 2830 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → (𝑧 = 𝑤 ↔ (𝑓𝑏) = 𝑤))
9593, 94imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9695ralbidv 3202 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑓𝑏) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9796, 2elrab2 3687 . . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐴)
10191, 99, 100rspcdva 3629 . . . . . . . . . . . . 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 864 . . . . . . . . . . . . 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 1422 . . . . . . . . . . 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 3196 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
108 2fveq3 6674 . . . . . . . . . 10 (𝑎 = 𝑏 → ((,)‘(𝑓𝑎)) = ((,)‘(𝑓𝑏)))
109108disjor 5043 . . . . . . . . 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 2826 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
11249, 110, 111uniiccmbl 24125 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) ∈ dom vol)
11345, 112eqeltrrd 2919 . . . . . 6 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ 𝐺) ∈ dom vol)
114113ex 413 . . . . 5 (𝜑 → (𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
115114exlimdv 1927 . . . 4 (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
116 nnenom 13343 . . . . . 6 ℕ ≈ ω
117 ensym 8552 . . . . . 6 (𝐺 ≈ ω → ω ≈ 𝐺)
118 entr 8555 . . . . . 6 ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
119116, 117, 118sylancr 587 . . . . 5 (𝐺 ≈ ω → ℕ ≈ 𝐺)
120 bren 8512 . . . . 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 10622 . . . . . . . . 9 ℝ ∈ V
124123, 123xpex 7469 . . . . . . . 8 (ℝ × ℝ) ∈ V
125124inex2 5219 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ∈ V
126125, 15ssexi 5223 . . . . . 6 ran 𝐹 ∈ V
127 ssdomg 8549 . . . . . 6 (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹𝐺 ≼ ran 𝐹))
128126, 12, 127mpsyl 68 . . . . 5 (𝜑𝐺 ≼ ran 𝐹)
129 omelon 9103 . . . . . . . 8 ω ∈ On
130 znnen 15560 . . . . . . . . . . . 12 ℤ ≈ ℕ
131130, 116entri 8557 . . . . . . . . . . 11 ℤ ≈ ω
132 nn0ennn 13342 . . . . . . . . . . . 12 0 ≈ ℕ
133132, 116entri 8557 . . . . . . . . . . 11 0 ≈ ω
134 xpen 8674 . . . . . . . . . . 11 ((ℤ ≈ ω ∧ ℕ0 ≈ ω) → (ℤ × ℕ0) ≈ (ω × ω))
135131, 133, 134mp2an 688 . . . . . . . . . 10 (ℤ × ℕ0) ≈ (ω × ω)
136 xpomen 9435 . . . . . . . . . 10 (ω × ω) ≈ ω
137135, 136entri 8557 . . . . . . . . 9 (ℤ × ℕ0) ≈ ω
138137ensymi 8553 . . . . . . . 8 ω ≈ (ℤ × ℕ0)
139 isnumi 9369 . . . . . . . 8 ((ω ∈ On ∧ ω ≈ (ℤ × ℕ0)) → (ℤ × ℕ0) ∈ dom card)
140129, 138, 139mp2an 688 . . . . . . 7 (ℤ × ℕ0) ∈ dom card
141 ffn 6513 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14213, 141ax-mp 5 . . . . . . . 8 𝐹 Fn (ℤ × ℕ0)
143 dffn4 6595 . . . . . . . 8 (𝐹 Fn (ℤ × ℕ0) ↔ 𝐹:(ℤ × ℕ0)–onto→ran 𝐹)
144142, 143mpbi 231 . . . . . . 7 𝐹:(ℤ × ℕ0)–onto→ran 𝐹
145 fodomnum 9477 . . . . . . 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 8562 . . . . . 6 ((ran 𝐹 ≼ (ℤ × ℕ0) ∧ (ℤ × ℕ0) ≈ ω) → ran 𝐹 ≼ ω)
148146, 137, 147mp2an 688 . . . . 5 ran 𝐹 ≼ ω
149 domtr 8556 . . . . 5 ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
150128, 148, 149sylancl 586 . . . 4 (𝜑𝐺 ≼ ω)
151 brdom2 8533 . . . 4 (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
152150, 151sylib 219 . . 3 (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
15337, 122, 152mpjaodan 954 . 2 (𝜑 ([,] “ 𝐺) ∈ dom vol)
1544, 153eqeltrd 2918 1 (𝜑 ([,] “ 𝐴) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3o 1080   = wceq 1530  wex 1773  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  cop 4570   cuni 4837   ciun 4917  Disj wdisj 5028   class class class wbr 5063   × cxp 5552  dom cdm 5554  ran crn 5555  cima 5557  ccom 5558  Oncon0 6190  Fun wfun 6348   Fn wfn 6349  wf 6350  1-1wf1 6351  ontowfo 6352  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7150  cmpo 7152  ωcom 7573  1st c1st 7683  2nd c2nd 7684  cen 8500  cdom 8501  csdm 8502  Fincfn 8503  cardccrd 9358  cr 10530  1c1 10532   + caddc 10534  *cxr 10668  cle 10670  cmin 10864   / cdiv 11291  cn 11632  2c2 11686  0cn0 11891  cz 11975  (,)cioo 12733  [,]cicc 12736  seqcseq 13364  cexp 13424  abscabs 14588  volcvol 23998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-omul 8103  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-acn 9365  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ioo 12737  df-ico 12739  df-icc 12740  df-fz 12888  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-rest 16691  df-topgen 16712  df-psmet 20472  df-xmet 20473  df-met 20474  df-bl 20475  df-mopn 20476  df-top 21437  df-topon 21454  df-bases 21489  df-cmp 21930  df-ovol 23999  df-vol 24000
This theorem is referenced by:  opnmbllem  24136
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