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 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)}
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)

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 23173 . 2 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
5 isfinite 8432 . . . 4 (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6 iccf 12143 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 ffun 5961 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
8 funiunfv 6410 . . . . . 6 (Fun [,] → 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺))
96, 7, 8mp2b 10 . . . . 5 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺)
10 simpr 476 . . . . . 6 ((𝜑𝐺 ∈ Fin) → 𝐺 ∈ Fin)
11 ssrab2 3650 . . . . . . . . . . . . . . . 16 {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} ⊆ 𝐴
122, 11eqsstri 3598 . . . . . . . . . . . . . . 15 𝐺𝐴
1312, 3syl5ss 3579 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ ran 𝐹)
141dyadf 23165 . . . . . . . . . . . . . . . 16 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
15 frn 5966 . . . . . . . . . . . . . . . 16 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
17 inss2 3796 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
1816, 17sstri 3577 . . . . . . . . . . . . . 14 ran 𝐹 ⊆ (ℝ × ℝ)
1913, 18syl6ss 3580 . . . . . . . . . . . . 13 (𝜑𝐺 ⊆ (ℝ × ℝ))
2019adantr 480 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
2120sselda 3568 . . . . . . . . . . 11 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 ∈ (ℝ × ℝ))
22 1st2nd2 7096 . . . . . . . . . . 11 (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2321, 22syl 17 . . . . . . . . . 10 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2423fveq2d 6107 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩))
25 df-ov 6552 . . . . . . . . 9 ((1st𝑛)[,](2nd𝑛)) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩)
2624, 25syl6eqr 2662 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ((1st𝑛)[,](2nd𝑛)))
27 xp1st 7089 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (1st𝑛) ∈ ℝ)
2821, 27syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (1st𝑛) ∈ ℝ)
29 xp2nd 7090 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (2nd𝑛) ∈ ℝ)
3021, 29syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (2nd𝑛) ∈ ℝ)
31 iccmbl 23141 . . . . . . . . 9 (((1st𝑛) ∈ ℝ ∧ (2nd𝑛) ∈ ℝ) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3228, 30, 31syl2anc 691 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3326, 32eqeltrd 2688 . . . . . . 7 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) ∈ dom vol)
3433ralrimiva 2949 . . . . . 6 ((𝜑𝐺 ∈ Fin) → ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
35 finiunmbl 23119 . . . . . 6 ((𝐺 ∈ Fin ∧ ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
3610, 34, 35syl2anc 691 . . . . 5 ((𝜑𝐺 ∈ Fin) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
379, 36syl5eqelr 2693 . . . 4 ((𝜑𝐺 ∈ Fin) → ([,] “ 𝐺) ∈ dom vol)
385, 37sylan2br 492 . . 3 ((𝜑𝐺 ≺ ω) → ([,] “ 𝐺) ∈ dom vol)
39 nnenom 12641 . . . . . . 7 ℕ ≈ ω
40 ensym 7891 . . . . . . 7 (𝐺 ≈ ω → ω ≈ 𝐺)
41 entr 7894 . . . . . . 7 ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
4239, 40, 41sylancr 694 . . . . . 6 (𝐺 ≈ ω → ℕ ≈ 𝐺)
43 bren 7850 . . . . . 6 (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
4442, 43sylib 207 . . . . 5 (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
45 rnco2 5559 . . . . . . . . . 10 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
46 f1ofo 6057 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–onto𝐺)
4746adantl 481 . . . . . . . . . . . 12 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–onto𝐺)
48 forn 6031 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐺 → ran 𝑓 = 𝐺)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran 𝑓 = 𝐺)
5049imaeq2d 5385 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
5145, 50syl5eq 2656 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
5251unieqd 4382 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
53 f1of 6050 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ⟶𝐺)
5413, 16syl6ss 3580 . . . . . . . . . 10 (𝜑𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
55 fss 5969 . . . . . . . . . 10 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5653, 54, 55syl2anr 494 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
57 fss 5969 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
5853, 13, 57syl2anr 494 . . . . . . . . . . . . . 14 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶ran 𝐹)
59 simpl 472 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
60 ffvelrn 6265 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑎 ∈ ℕ) → (𝑓𝑎) ∈ ran 𝐹)
6158, 59, 60syl2an 493 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ ran 𝐹)
62 simpr 476 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
63 ffvelrn 6265 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑏 ∈ ℕ) → (𝑓𝑏) ∈ ran 𝐹)
6458, 62, 63syl2an 493 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ ran 𝐹)
651dyaddisj 23170 . . . . . . . . . . . . 13 (((𝑓𝑎) ∈ ran 𝐹 ∧ (𝑓𝑏) ∈ ran 𝐹) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
6661, 64, 65syl2anc 691 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
6753adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶𝐺)
68 ffvelrn 6265 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑏 ∈ ℕ) → (𝑓𝑏) ∈ 𝐺)
6967, 62, 68syl2an 493 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐺)
7012, 69sseldi 3566 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐴)
71 ffvelrn 6265 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑎 ∈ ℕ) → (𝑓𝑎) ∈ 𝐺)
7267, 59, 71syl2an 493 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐺)
73 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑎) → ([,]‘𝑧) = ([,]‘(𝑓𝑎)))
7473sseq1d 3595 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤)))
75 eqeq1 2614 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (𝑧 = 𝑤 ↔ (𝑓𝑎) = 𝑤))
7674, 75imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7776ralbidv 2969 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑎) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7877, 2elrab2 3333 . . . . . . . . . . . . . . . . 17 ((𝑓𝑎) ∈ 𝐺 ↔ ((𝑓𝑎) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7978simprbi 479 . . . . . . . . . . . . . . . 16 ((𝑓𝑎) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
8072, 79syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤))
81 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑏) → ([,]‘𝑤) = ([,]‘(𝑓𝑏)))
8281sseq2d 3596 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑏) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏))))
83 eqeq2 2621 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑏) → ((𝑓𝑎) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
8482, 83imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
8584rspcv 3278 . . . . . . . . . . . . . . 15 ((𝑓𝑏) ∈ 𝐴 → (∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
8670, 80, 85sylc 63 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏)))
87 f1of1 6049 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–1-1𝐺)
8887adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–1-1𝐺)
89 f1fveq 6420 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
9088, 89sylan 487 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
91 orc 399 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
9290, 91syl6bi 242 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
9386, 92syld 46 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
9412, 72sseldi 3566 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐴)
95 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑏) → ([,]‘𝑧) = ([,]‘(𝑓𝑏)))
9695sseq1d 3595 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤)))
97 eqeq1 2614 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (𝑧 = 𝑤 ↔ (𝑓𝑏) = 𝑤))
9896, 97imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9998ralbidv 2969 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑏) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
10099, 2elrab2 3333 . . . . . . . . . . . . . . . . 17 ((𝑓𝑏) ∈ 𝐺 ↔ ((𝑓𝑏) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
101100simprbi 479 . . . . . . . . . . . . . . . 16 ((𝑓𝑏) ∈ 𝐺 → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
10269, 101syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤))
103 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑎) → ([,]‘𝑤) = ([,]‘(𝑓𝑎)))
104103sseq2d 3596 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎))))
105 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑏) = (𝑓𝑎)))
106 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑏) = (𝑓𝑎) ↔ (𝑓𝑎) = (𝑓𝑏))
107105, 106syl6bb 275 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
108104, 107imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ((([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
109108rspcv 3278 . . . . . . . . . . . . . . 15 ((𝑓𝑎) ∈ 𝐴 → (∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
11094, 102, 109sylc 63 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏)))
111110, 92syld 46 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
112 olc 398 . . . . . . . . . . . . . 14 ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
113112a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅ → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
11493, 111, 1133jaod 1384 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
11566, 114mpd 15 . . . . . . . . . . 11 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
116115ralrimivva 2954 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
117 fveq2 6103 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑓𝑎) = (𝑓𝑏))
118117fveq2d 6107 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((,)‘(𝑓𝑎)) = ((,)‘(𝑓𝑏)))
119118disjor 4567 . . . . . . . . . 10 (Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
120116, 119sylibr 223 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)))
121 eqid 2610 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
12256, 120, 121uniiccmbl 23164 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) ∈ dom vol)
12352, 122eqeltrrd 2689 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ 𝐺) ∈ dom vol)
124123ex 449 . . . . . 6 (𝜑 → (𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
125124exlimdv 1848 . . . . 5 (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
12644, 125syl5 33 . . . 4 (𝜑 → (𝐺 ≈ ω → ([,] “ 𝐺) ∈ dom vol))
127126imp 444 . . 3 ((𝜑𝐺 ≈ ω) → ([,] “ 𝐺) ∈ dom vol)
128 reex 9906 . . . . . . . . 9 ℝ ∈ V
129128, 128xpex 6860 . . . . . . . 8 (ℝ × ℝ) ∈ V
130129inex2 4728 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ∈ V
131130, 16ssexi 4731 . . . . . 6 ran 𝐹 ∈ V
132 ssdomg 7887 . . . . . 6 (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹𝐺 ≼ ran 𝐹))
133131, 13, 132mpsyl 66 . . . . 5 (𝜑𝐺 ≼ ran 𝐹)
134 omelon 8426 . . . . . . . 8 ω ∈ On
135 znnen 14780 . . . . . . . . . . . 12 ℤ ≈ ℕ
136135, 39entri 7896 . . . . . . . . . . 11 ℤ ≈ ω
137 nn0ennn 12640 . . . . . . . . . . . 12 0 ≈ ℕ
138137, 39entri 7896 . . . . . . . . . . 11 0 ≈ ω
139 xpen 8008 . . . . . . . . . . 11 ((ℤ ≈ ω ∧ ℕ0 ≈ ω) → (ℤ × ℕ0) ≈ (ω × ω))
140136, 138, 139mp2an 704 . . . . . . . . . 10 (ℤ × ℕ0) ≈ (ω × ω)
141 xpomen 8721 . . . . . . . . . 10 (ω × ω) ≈ ω
142140, 141entri 7896 . . . . . . . . 9 (ℤ × ℕ0) ≈ ω
143142ensymi 7892 . . . . . . . 8 ω ≈ (ℤ × ℕ0)
144 isnumi 8655 . . . . . . . 8 ((ω ∈ On ∧ ω ≈ (ℤ × ℕ0)) → (ℤ × ℕ0) ∈ dom card)
145134, 143, 144mp2an 704 . . . . . . 7 (ℤ × ℕ0) ∈ dom card
146 ffn 5958 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14714, 146ax-mp 5 . . . . . . . 8 𝐹 Fn (ℤ × ℕ0)
148 dffn4 6034 . . . . . . . 8 (𝐹 Fn (ℤ × ℕ0) ↔ 𝐹:(ℤ × ℕ0)–onto→ran 𝐹)
149147, 148mpbi 219 . . . . . . 7 𝐹:(ℤ × ℕ0)–onto→ran 𝐹
150 fodomnum 8763 . . . . . . 7 ((ℤ × ℕ0) ∈ dom card → (𝐹:(ℤ × ℕ0)–onto→ran 𝐹 → ran 𝐹 ≼ (ℤ × ℕ0)))
151145, 149, 150mp2 9 . . . . . 6 ran 𝐹 ≼ (ℤ × ℕ0)
152 domentr 7901 . . . . . 6 ((ran 𝐹 ≼ (ℤ × ℕ0) ∧ (ℤ × ℕ0) ≈ ω) → ran 𝐹 ≼ ω)
153151, 142, 152mp2an 704 . . . . 5 ran 𝐹 ≼ ω
154 domtr 7895 . . . . 5 ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
155133, 153, 154sylancl 693 . . . 4 (𝜑𝐺 ≼ ω)
156 brdom2 7871 . . . 4 (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
157155, 156sylib 207 . . 3 (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
15838, 127, 157mpjaodan 823 . 2 (𝜑 ([,] “ 𝐺) ∈ dom vol)
1594, 158eqeltrd 2688 1 (𝜑 ([,] “ 𝐴) ∈ dom vol)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  2nd c2nd 7058   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  cardccrd 8644  ℝcr 9814  1c1 9816   + caddc 9818  ℝ*cxr 9952   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  (,)cioo 12046  [,]cicc 12049  seqcseq 12663  ↑cexp 12722  abscabs 13822  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041 This theorem is referenced by:  opnmbllem  23175
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