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Theorem dyadmbl 23287
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 23286 . 2 (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))
5 isfinite 8500 . . . 4 (𝐺 ∈ Fin ↔ 𝐺 ≺ ω)
6 iccf 12221 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 ffun 6010 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
8 funiunfv 6466 . . . . . 6 (Fun [,] → 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺))
96, 7, 8mp2b 10 . . . . 5 𝑛𝐺 ([,]‘𝑛) = ([,] “ 𝐺)
10 simpr 477 . . . . . 6 ((𝜑𝐺 ∈ Fin) → 𝐺 ∈ Fin)
11 ssrab2 3671 . . . . . . . . . . . . . . . 16 {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} ⊆ 𝐴
122, 11eqsstri 3619 . . . . . . . . . . . . . . 15 𝐺𝐴
1312, 3syl5ss 3598 . . . . . . . . . . . . . 14 (𝜑𝐺 ⊆ ran 𝐹)
141dyadf 23278 . . . . . . . . . . . . . . . 16 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
15 frn 6015 . . . . . . . . . . . . . . . 16 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15 ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
17 inss2 3817 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
1816, 17sstri 3596 . . . . . . . . . . . . . 14 ran 𝐹 ⊆ (ℝ × ℝ)
1913, 18syl6ss 3599 . . . . . . . . . . . . 13 (𝜑𝐺 ⊆ (ℝ × ℝ))
2019adantr 481 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ Fin) → 𝐺 ⊆ (ℝ × ℝ))
2120sselda 3587 . . . . . . . . . . 11 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 ∈ (ℝ × ℝ))
22 1st2nd2 7157 . . . . . . . . . . 11 (𝑛 ∈ (ℝ × ℝ) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2321, 22syl 17 . . . . . . . . . 10 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
2423fveq2d 6157 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩))
25 df-ov 6613 . . . . . . . . 9 ((1st𝑛)[,](2nd𝑛)) = ([,]‘⟨(1st𝑛), (2nd𝑛)⟩)
2624, 25syl6eqr 2673 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) = ((1st𝑛)[,](2nd𝑛)))
27 xp1st 7150 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (1st𝑛) ∈ ℝ)
2821, 27syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (1st𝑛) ∈ ℝ)
29 xp2nd 7151 . . . . . . . . . 10 (𝑛 ∈ (ℝ × ℝ) → (2nd𝑛) ∈ ℝ)
3021, 29syl 17 . . . . . . . . 9 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → (2nd𝑛) ∈ ℝ)
31 iccmbl 23253 . . . . . . . . 9 (((1st𝑛) ∈ ℝ ∧ (2nd𝑛) ∈ ℝ) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3228, 30, 31syl2anc 692 . . . . . . . 8 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ((1st𝑛)[,](2nd𝑛)) ∈ dom vol)
3326, 32eqeltrd 2698 . . . . . . 7 (((𝜑𝐺 ∈ Fin) ∧ 𝑛𝐺) → ([,]‘𝑛) ∈ dom vol)
3433ralrimiva 2961 . . . . . 6 ((𝜑𝐺 ∈ Fin) → ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
35 finiunmbl 23231 . . . . . 6 ((𝐺 ∈ Fin ∧ ∀𝑛𝐺 ([,]‘𝑛) ∈ dom vol) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
3610, 34, 35syl2anc 692 . . . . 5 ((𝜑𝐺 ∈ Fin) → 𝑛𝐺 ([,]‘𝑛) ∈ dom vol)
379, 36syl5eqelr 2703 . . . 4 ((𝜑𝐺 ∈ Fin) → ([,] “ 𝐺) ∈ dom vol)
385, 37sylan2br 493 . . 3 ((𝜑𝐺 ≺ ω) → ([,] “ 𝐺) ∈ dom vol)
39 nnenom 12726 . . . . . . 7 ℕ ≈ ω
40 ensym 7956 . . . . . . 7 (𝐺 ≈ ω → ω ≈ 𝐺)
41 entr 7959 . . . . . . 7 ((ℕ ≈ ω ∧ ω ≈ 𝐺) → ℕ ≈ 𝐺)
4239, 40, 41sylancr 694 . . . . . 6 (𝐺 ≈ ω → ℕ ≈ 𝐺)
43 bren 7915 . . . . . 6 (ℕ ≈ 𝐺 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
4442, 43sylib 208 . . . . 5 (𝐺 ≈ ω → ∃𝑓 𝑓:ℕ–1-1-onto𝐺)
45 rnco2 5606 . . . . . . . . . 10 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
46 f1ofo 6106 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–onto𝐺)
4746adantl 482 . . . . . . . . . . . 12 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–onto𝐺)
48 forn 6080 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐺 → ran 𝑓 = 𝐺)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran 𝑓 = 𝐺)
5049imaeq2d 5430 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ ran 𝑓) = ([,] “ 𝐺))
5145, 50syl5eq 2667 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
5251unieqd 4417 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) = ([,] “ 𝐺))
53 f1of 6099 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ⟶𝐺)
5413, 16syl6ss 3599 . . . . . . . . . 10 (𝜑𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ)))
55 fss 6018 . . . . . . . . . 10 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5653, 54, 55syl2anr 495 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
57 fss 6018 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶𝐺𝐺 ⊆ ran 𝐹) → 𝑓:ℕ⟶ran 𝐹)
5853, 13, 57syl2anr 495 . . . . . . . . . . . . . 14 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶ran 𝐹)
59 simpl 473 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑎 ∈ ℕ)
60 ffvelrn 6318 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑎 ∈ ℕ) → (𝑓𝑎) ∈ ran 𝐹)
6158, 59, 60syl2an 494 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ ran 𝐹)
62 simpr 477 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → 𝑏 ∈ ℕ)
63 ffvelrn 6318 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶ran 𝐹𝑏 ∈ ℕ) → (𝑓𝑏) ∈ ran 𝐹)
6458, 62, 63syl2an 494 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ ran 𝐹)
651dyaddisj 23283 . . . . . . . . . . . . 13 (((𝑓𝑎) ∈ ran 𝐹 ∧ (𝑓𝑏) ∈ ran 𝐹) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
6661, 64, 65syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) ∨ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
6753adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ⟶𝐺)
68 ffvelrn 6318 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑏 ∈ ℕ) → (𝑓𝑏) ∈ 𝐺)
6967, 62, 68syl2an 494 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐺)
7012, 69sseldi 3585 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑏) ∈ 𝐴)
71 ffvelrn 6318 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶𝐺𝑎 ∈ ℕ) → (𝑓𝑎) ∈ 𝐺)
7267, 59, 71syl2an 494 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐺)
73 fveq2 6153 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑎) → ([,]‘𝑧) = ([,]‘(𝑓𝑎)))
7473sseq1d 3616 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤)))
75 eqeq1 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑎) → (𝑧 = 𝑤 ↔ (𝑓𝑎) = 𝑤))
7674, 75imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑎) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7776ralbidv 2981 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑎) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7877, 2elrab2 3352 . . . . . . . . . . . . . . . . 17 ((𝑓𝑎) ∈ 𝐺 ↔ ((𝑓𝑎) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤)))
7978simprbi 480 . . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑏) → ([,]‘𝑤) = ([,]‘(𝑓𝑏)))
8281sseq2d 3617 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑏) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏))))
83 eqeq2 2632 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑏) → ((𝑓𝑎) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
8482, 83imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑏) → ((([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) ↔ (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
8584rspcv 3294 . . . . . . . . . . . . . . 15 ((𝑓𝑏) ∈ 𝐴 → (∀𝑤𝐴 (([,]‘(𝑓𝑎)) ⊆ ([,]‘𝑤) → (𝑓𝑎) = 𝑤) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏))))
8670, 80, 85sylc 65 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑓𝑎) = (𝑓𝑏)))
87 f1of1 6098 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐺𝑓:ℕ–1-1𝐺)
8887adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑓:ℕ–1-1-onto𝐺) → 𝑓:ℕ–1-1𝐺)
89 f1fveq 6479 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1𝐺 ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
9088, 89sylan 488 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) ↔ 𝑎 = 𝑏))
91 orc 400 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
9290, 91syl6bi 243 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → ((𝑓𝑎) = (𝑓𝑏) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
9386, 92syld 47 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑎)) ⊆ ([,]‘(𝑓𝑏)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
9412, 72sseldi 3585 . . . . . . . . . . . . . . 15 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (𝑓𝑎) ∈ 𝐴)
95 fveq2 6153 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑓𝑏) → ([,]‘𝑧) = ([,]‘(𝑓𝑏)))
9695sseq1d 3616 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤)))
97 eqeq1 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑓𝑏) → (𝑧 = 𝑤 ↔ (𝑓𝑏) = 𝑤))
9896, 97imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑓𝑏) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
9998ralbidv 2981 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑏) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
10099, 2elrab2 3352 . . . . . . . . . . . . . . . . 17 ((𝑓𝑏) ∈ 𝐺 ↔ ((𝑓𝑏) ∈ 𝐴 ∧ ∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤)))
101100simprbi 480 . . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑎) → ([,]‘𝑤) = ([,]‘(𝑓𝑎)))
104103sseq2d 3617 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎))))
105 eqeq2 2632 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑏) = (𝑓𝑎)))
106 eqcom 2628 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑏) = (𝑓𝑎) ↔ (𝑓𝑎) = (𝑓𝑏))
107105, 106syl6bb 276 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑎) → ((𝑓𝑏) = 𝑤 ↔ (𝑓𝑎) = (𝑓𝑏)))
108104, 107imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑎) → ((([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) ↔ (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
109108rspcv 3294 . . . . . . . . . . . . . . 15 ((𝑓𝑎) ∈ 𝐴 → (∀𝑤𝐴 (([,]‘(𝑓𝑏)) ⊆ ([,]‘𝑤) → (𝑓𝑏) = 𝑤) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏))))
11094, 102, 109sylc 65 . . . . . . . . . . . . . 14 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑓𝑎) = (𝑓𝑏)))
111110, 92syld 47 . . . . . . . . . . . . 13 (((𝜑𝑓:ℕ–1-1-onto𝐺) ∧ (𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ)) → (([,]‘(𝑓𝑏)) ⊆ ([,]‘(𝑓𝑎)) → (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅)))
112 olc 399 . . . . . . . . . . . . . 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 1389 . . . . . . . . . . . 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 2966 . . . . . . . . . 10 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
117 fveq2 6153 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑓𝑎) = (𝑓𝑏))
118117fveq2d 6157 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((,)‘(𝑓𝑎)) = ((,)‘(𝑓𝑏)))
119118disjor 4602 . . . . . . . . . 10 (Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ (𝑎 = 𝑏 ∨ (((,)‘(𝑓𝑎)) ∩ ((,)‘(𝑓𝑏))) = ∅))
120116, 119sylibr 224 . . . . . . . . 9 ((𝜑𝑓:ℕ–1-1-onto𝐺) → Disj 𝑎 ∈ ℕ ((,)‘(𝑓𝑎)))
121 eqid 2621 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
12256, 120, 121uniiccmbl 23277 . . . . . . . 8 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ran ([,] ∘ 𝑓) ∈ dom vol)
12352, 122eqeltrrd 2699 . . . . . . 7 ((𝜑𝑓:ℕ–1-1-onto𝐺) → ([,] “ 𝐺) ∈ dom vol)
124123ex 450 . . . . . 6 (𝜑 → (𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
125124exlimdv 1858 . . . . 5 (𝜑 → (∃𝑓 𝑓:ℕ–1-1-onto𝐺 ([,] “ 𝐺) ∈ dom vol))
12644, 125syl5 34 . . . 4 (𝜑 → (𝐺 ≈ ω → ([,] “ 𝐺) ∈ dom vol))
127126imp 445 . . 3 ((𝜑𝐺 ≈ ω) → ([,] “ 𝐺) ∈ dom vol)
128 reex 9978 . . . . . . . . 9 ℝ ∈ V
129128, 128xpex 6922 . . . . . . . 8 (ℝ × ℝ) ∈ V
130129inex2 4765 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ∈ V
131130, 16ssexi 4768 . . . . . 6 ran 𝐹 ∈ V
132 ssdomg 7952 . . . . . 6 (ran 𝐹 ∈ V → (𝐺 ⊆ ran 𝐹𝐺 ≼ ran 𝐹))
133131, 13, 132mpsyl 68 . . . . 5 (𝜑𝐺 ≼ ran 𝐹)
134 omelon 8494 . . . . . . . 8 ω ∈ On
135 znnen 14873 . . . . . . . . . . . 12 ℤ ≈ ℕ
136135, 39entri 7961 . . . . . . . . . . 11 ℤ ≈ ω
137 nn0ennn 12725 . . . . . . . . . . . 12 0 ≈ ℕ
138137, 39entri 7961 . . . . . . . . . . 11 0 ≈ ω
139 xpen 8074 . . . . . . . . . . 11 ((ℤ ≈ ω ∧ ℕ0 ≈ ω) → (ℤ × ℕ0) ≈ (ω × ω))
140136, 138, 139mp2an 707 . . . . . . . . . 10 (ℤ × ℕ0) ≈ (ω × ω)
141 xpomen 8789 . . . . . . . . . 10 (ω × ω) ≈ ω
142140, 141entri 7961 . . . . . . . . 9 (ℤ × ℕ0) ≈ ω
143142ensymi 7957 . . . . . . . 8 ω ≈ (ℤ × ℕ0)
144 isnumi 8723 . . . . . . . 8 ((ω ∈ On ∧ ω ≈ (ℤ × ℕ0)) → (ℤ × ℕ0) ∈ dom card)
145134, 143, 144mp2an 707 . . . . . . 7 (ℤ × ℕ0) ∈ dom card
146 ffn 6007 . . . . . . . . 9 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14714, 146ax-mp 5 . . . . . . . 8 𝐹 Fn (ℤ × ℕ0)
148 dffn4 6083 . . . . . . . 8 (𝐹 Fn (ℤ × ℕ0) ↔ 𝐹:(ℤ × ℕ0)–onto→ran 𝐹)
149147, 148mpbi 220 . . . . . . 7 𝐹:(ℤ × ℕ0)–onto→ran 𝐹
150 fodomnum 8831 . . . . . . 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 7966 . . . . . 6 ((ran 𝐹 ≼ (ℤ × ℕ0) ∧ (ℤ × ℕ0) ≈ ω) → ran 𝐹 ≼ ω)
153151, 142, 152mp2an 707 . . . . 5 ran 𝐹 ≼ ω
154 domtr 7960 . . . . 5 ((𝐺 ≼ ran 𝐹 ∧ ran 𝐹 ≼ ω) → 𝐺 ≼ ω)
155133, 153, 154sylancl 693 . . . 4 (𝜑𝐺 ≼ ω)
156 brdom2 7936 . . . 4 (𝐺 ≼ ω ↔ (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
157155, 156sylib 208 . . 3 (𝜑 → (𝐺 ≺ ω ∨ 𝐺 ≈ ω))
15838, 127, 157mpjaodan 826 . 2 (𝜑 ([,] “ 𝐺) ∈ dom vol)
1594, 158eqeltrd 2698 1 (𝜑 ([,] “ 𝐴) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3o 1035   = wceq 1480  wex 1701  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  cop 4159   cuni 4407   ciun 4490  Disj wdisj 4588   class class class wbr 4618   × cxp 5077  dom cdm 5079  ran crn 5080  cima 5082  ccom 5083  Oncon0 5687  Fun wfun 5846   Fn wfn 5847  wf 5848  1-1wf1 5849  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  ωcom 7019  1st c1st 7118  2nd c2nd 7119  cen 7903  cdom 7904  csdm 7905  Fincfn 7906  cardccrd 8712  cr 9886  1c1 9888   + caddc 9890  *cxr 10024  cle 10026  cmin 10217   / cdiv 10635  cn 10971  2c2 11021  0cn0 11243  cz 11328  (,)cioo 12124  [,]cicc 12127  seqcseq 12748  cexp 12807  abscabs 13915  volcvol 23151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-acn 8719  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-n0 11244  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ioo 12128  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-rlim 14161  df-sum 14358  df-rest 16011  df-topgen 16032  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-top 20627  df-topon 20644  df-bases 20670  df-cmp 21109  df-ovol 23152  df-vol 23153
This theorem is referenced by:  opnmbllem  23288
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