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Theorem uniiccdif 23566
Description: A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypothesis
Ref Expression
uniioombl.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
Assertion
Ref Expression
uniiccdif (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))

Proof of Theorem uniiccdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssun1 3927 . . 3 ran ((,) ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 23454 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
42, 3sylan 561 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
5 rexr 10287 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ∈ ℝ → (1st ‘(𝐹𝑥)) ∈ ℝ*)
6 rexr 10287 . . . . . . . 8 ((2nd ‘(𝐹𝑥)) ∈ ℝ → (2nd ‘(𝐹𝑥)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)) → (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)))
8 prunioo 12508 . . . . . . . 8 (((1st ‘(𝐹𝑥)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ* ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))) → (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
95, 6, 7, 8syl3an 1163 . . . . . . 7 (((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))) → (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
104, 9syl 17 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
11 fvco3 6417 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
122, 11sylan 561 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
13 inss2 3982 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
142ffvelrnda 6502 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1513, 14sseldi 3750 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
16 1st2nd2 7354 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1715, 16syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1817fveq2d 6336 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
19 df-ov 6796 . . . . . . . . 9 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2018, 19syl6eqr 2823 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2112, 20eqtrd 2805 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
22 df-pr 4319 . . . . . . . 8 {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})
23 fvco3 6417 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
242, 23sylan 561 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
25 fvco3 6417 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
262, 25sylan 561 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
2724, 26preq12d 4412 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2822, 27syl5eqr 2819 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2921, 28uneq12d 3919 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}))
30 fvco3 6417 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
312, 30sylan 561 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3217fveq2d 6336 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
33 df-ov 6796 . . . . . . . 8 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3432, 33syl6eqr 2823 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3531, 34eqtrd 2805 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3610, 29, 353eqtr4rd 2816 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
3736iuneq2dv 4676 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
38 iccf 12478 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
39 ffn 6185 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
4038, 39ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
41 rexpssxrxp 10286 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4213, 41sstri 3761 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43 fss 6196 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
442, 42, 43sylancl 566 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
45 fnfco 6209 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4640, 44, 45sylancr 567 . . . . 5 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47 fniunfv 6648 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
49 iunun 4738 . . . . 5 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}))
50 ioof 12477 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51 ffn 6185 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* × ℝ*)
53 fnfco 6209 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
5452, 44, 53sylancr 567 . . . . . . 7 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55 fniunfv 6648 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
57 iunun 4738 . . . . . . 7 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
58 fo1st 7335 . . . . . . . . . . . . . 14 1st :V–onto→V
59 fofn 6258 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 3774 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62 fss 6196 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
632, 61, 62sylancl 566 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶V)
64 fnfco 6209 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st𝐹) Fn ℕ)
6560, 63, 64sylancr 567 . . . . . . . . . . . 12 (𝜑 → (1st𝐹) Fn ℕ)
66 fnfun 6128 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → Fun (1st𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (𝜑 → Fun (1st𝐹))
68 fndm 6130 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → dom (1st𝐹) = ℕ)
69 eqimss2 3807 . . . . . . . . . . . 12 (dom (1st𝐹) = ℕ → ℕ ⊆ dom (1st𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (1st𝐹))
71 dfimafn2 6388 . . . . . . . . . . 11 ((Fun (1st𝐹) ∧ ℕ ⊆ dom (1st𝐹)) → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
7267, 70, 71syl2anc 565 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
73 fnima 6150 . . . . . . . . . . 11 ((1st𝐹) Fn ℕ → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7465, 73syl 17 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7572, 74eqtr3d 2807 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = ran (1st𝐹))
76 rnco2 5786 . . . . . . . . 9 ran (1st𝐹) = (1st “ ran 𝐹)
7775, 76syl6eq 2821 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = (1st “ ran 𝐹))
78 fo2nd 7336 . . . . . . . . . . . . . 14 2nd :V–onto→V
79 fofn 6258 . . . . . . . . . . . . . 14 (2nd :V–onto→V → 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6209 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd𝐹) Fn ℕ)
8280, 63, 81sylancr 567 . . . . . . . . . . . 12 (𝜑 → (2nd𝐹) Fn ℕ)
83 fnfun 6128 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → Fun (2nd𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd𝐹))
85 fndm 6130 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → dom (2nd𝐹) = ℕ)
86 eqimss2 3807 . . . . . . . . . . . 12 (dom (2nd𝐹) = ℕ → ℕ ⊆ dom (2nd𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (2nd𝐹))
88 dfimafn2 6388 . . . . . . . . . . 11 ((Fun (2nd𝐹) ∧ ℕ ⊆ dom (2nd𝐹)) → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
8984, 87, 88syl2anc 565 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
90 fnima 6150 . . . . . . . . . . 11 ((2nd𝐹) Fn ℕ → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9182, 90syl 17 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9289, 91eqtr3d 2807 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = ran (2nd𝐹))
93 rnco2 5786 . . . . . . . . 9 ran (2nd𝐹) = (2nd “ ran 𝐹)
9492, 93syl6eq 2821 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = (2nd “ ran 𝐹))
9577, 94uneq12d 3919 . . . . . . 7 (𝜑 → ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9657, 95syl5eq 2817 . . . . . 6 (𝜑 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9756, 96uneq12d 3919 . . . . 5 (𝜑 → ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9849, 97syl5eq 2817 . . . 4 (𝜑 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9937, 48, 983eqtr3d 2813 . . 3 (𝜑 ran ([,] ∘ 𝐹) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
1001, 99syl5sseqr 3803 . 2 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
101 ovolficcss 23457 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
1022, 101syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
103102ssdifssd 3899 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104 omelon 8707 . . . . . . . . . . 11 ω ∈ On
105 nnenom 12987 . . . . . . . . . . . 12 ℕ ≈ ω
106105ensymi 8159 . . . . . . . . . . 11 ω ≈ ℕ
107 isnumi 8972 . . . . . . . . . . 11 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108104, 106, 107mp2an 664 . . . . . . . . . 10 ℕ ∈ dom card
109 fofun 6257 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 3774 . . . . . . . . . . . . 13 ran 𝐹 ⊆ V
112 fof 6256 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
114113fdmi 6192 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtr4i 3787 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 1st
116 fores 6265 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹))
117110, 115, 116mp2an 664 . . . . . . . . . . 11 (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹)
118 ffn 6185 . . . . . . . . . . . . 13 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
1192, 118syl 17 . . . . . . . . . . . 12 (𝜑𝐹 Fn ℕ)
120 dffn4 6262 . . . . . . . . . . . 12 (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
121119, 120sylib 208 . . . . . . . . . . 11 (𝜑𝐹:ℕ–onto→ran 𝐹)
122 foco 6266 . . . . . . . . . . 11 (((1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123117, 121, 122sylancr 567 . . . . . . . . . 10 (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
124 fodomnum 9080 . . . . . . . . . 10 (ℕ ∈ dom card → (((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹) → (1st “ ran 𝐹) ≼ ℕ))
125108, 123, 124mpsyl 68 . . . . . . . . 9 (𝜑 → (1st “ ran 𝐹) ≼ ℕ)
126 domentr 8168 . . . . . . . . 9 (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
127125, 105, 126sylancl 566 . . . . . . . 8 (𝜑 → (1st “ ran 𝐹) ≼ ω)
128 fofun 6257 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
12978, 128ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
130 fof 6256 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
13178, 130ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
132131fdmi 6192 . . . . . . . . . . . . 13 dom 2nd = V
133111, 132sseqtr4i 3787 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 2nd
134 fores 6265 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹))
135129, 133, 134mp2an 664 . . . . . . . . . . 11 (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹)
136 foco 6266 . . . . . . . . . . 11 (((2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137135, 121, 136sylancr 567 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
138 fodomnum 9080 . . . . . . . . . 10 (ℕ ∈ dom card → (((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹) → (2nd “ ran 𝐹) ≼ ℕ))
139108, 137, 138mpsyl 68 . . . . . . . . 9 (𝜑 → (2nd “ ran 𝐹) ≼ ℕ)
140 domentr 8168 . . . . . . . . 9 (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
141139, 105, 140sylancl 566 . . . . . . . 8 (𝜑 → (2nd “ ran 𝐹) ≼ ω)
142 unctb 9229 . . . . . . . 8 (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143127, 141, 142syl2anc 565 . . . . . . 7 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
144 reldom 8115 . . . . . . . 8 Rel ≼
145144brrelexi 5298 . . . . . . 7 (((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
146143, 145syl 17 . . . . . 6 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
147 ssid 3773 . . . . . . . 8 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
148147, 99syl5sseq 3802 . . . . . . 7 (𝜑 ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
149 ssundif 4194 . . . . . . 7 ( ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))) ↔ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
150148, 149sylib 208 . . . . . 6 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
151 ssdomg 8155 . . . . . 6 (((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V → (( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
152146, 150, 151sylc 65 . . . . 5 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
153 domtr 8162 . . . . 5 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∧ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
154152, 143, 153syl2anc 565 . . . 4 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
155 domentr 8168 . . . 4 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
156154, 106, 155sylancl 566 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
157 ovolctb2 23480 . . 3 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
158103, 156, 157syl2anc 565 . 2 (𝜑 → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
159100, 158jca 495 1 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  𝒫 cpw 4297  {csn 4316  {cpr 4318  cop 4322   cuni 4574   ciun 4654   class class class wbr 4786   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  ccom 5253  Oncon0 5866  Fun wfun 6025   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  (class class class)co 6793  ωcom 7212  1st c1st 7313  2nd c2nd 7314  cen 8106  cdom 8107  cardccrd 8961  cr 10137  0cc0 10138  *cxr 10275  cle 10277  cn 11222  (,)cioo 12380  [,]cicc 12383  vol*covol 23450
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 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  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-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 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xadd 12152  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-xmet 19954  df-met 19955  df-ovol 23452
This theorem is referenced by:  uniioombllem3  23573  uniioombllem4  23574  uniioombllem5  23575  uniiccmbl  23578
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