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Theorem uniiccdif 23269
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 3759 . . 3 ran ((,) ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 23158 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
42, 3sylan 488 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
5 rexr 10037 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ∈ ℝ → (1st ‘(𝐹𝑥)) ∈ ℝ*)
6 rexr 10037 . . . . . . . 8 ((2nd ‘(𝐹𝑥)) ∈ ℝ → (2nd ‘(𝐹𝑥)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)) → (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)))
8 prunioo 12251 . . . . . . . 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 1365 . . . . . . 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 6237 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
122, 11sylan 488 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
13 inss2 3817 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
142ffvelrnda 6320 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1513, 14sseldi 3585 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
16 1st2nd2 7157 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1715, 16syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1817fveq2d 6157 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
19 df-ov 6613 . . . . . . . . 9 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2018, 19syl6eqr 2673 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2112, 20eqtrd 2655 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
22 df-pr 4156 . . . . . . . 8 {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})
23 fvco3 6237 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
242, 23sylan 488 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
25 fvco3 6237 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
262, 25sylan 488 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
2724, 26preq12d 4251 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2822, 27syl5eqr 2669 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2921, 28uneq12d 3751 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}))
30 fvco3 6237 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
312, 30sylan 488 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3217fveq2d 6157 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
33 df-ov 6613 . . . . . . . 8 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3432, 33syl6eqr 2673 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3531, 34eqtrd 2655 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3610, 29, 353eqtr4rd 2666 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
3736iuneq2dv 4513 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
38 iccf 12222 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
39 ffn 6007 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
4038, 39ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
41 rexpssxrxp 10036 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4213, 41sstri 3596 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43 fss 6018 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
442, 42, 43sylancl 693 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
45 fnfco 6031 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4640, 44, 45sylancr 694 . . . . 5 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47 fniunfv 6465 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
49 iunun 4575 . . . . 5 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}))
50 ioof 12221 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51 ffn 6007 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* × ℝ*)
53 fnfco 6031 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
5452, 44, 53sylancr 694 . . . . . . 7 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55 fniunfv 6465 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
57 iunun 4575 . . . . . . 7 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
58 fo1st 7140 . . . . . . . . . . . . . 14 1st :V–onto→V
59 fofn 6079 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 3609 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62 fss 6018 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
632, 61, 62sylancl 693 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶V)
64 fnfco 6031 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st𝐹) Fn ℕ)
6560, 63, 64sylancr 694 . . . . . . . . . . . 12 (𝜑 → (1st𝐹) Fn ℕ)
66 fnfun 5951 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → Fun (1st𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (𝜑 → Fun (1st𝐹))
68 fndm 5953 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → dom (1st𝐹) = ℕ)
69 eqimss2 3642 . . . . . . . . . . . 12 (dom (1st𝐹) = ℕ → ℕ ⊆ dom (1st𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (1st𝐹))
71 dfimafn2 6208 . . . . . . . . . . 11 ((Fun (1st𝐹) ∧ ℕ ⊆ dom (1st𝐹)) → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
7267, 70, 71syl2anc 692 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
73 fnima 5972 . . . . . . . . . . 11 ((1st𝐹) Fn ℕ → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7465, 73syl 17 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7572, 74eqtr3d 2657 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = ran (1st𝐹))
76 rnco2 5606 . . . . . . . . 9 ran (1st𝐹) = (1st “ ran 𝐹)
7775, 76syl6eq 2671 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = (1st “ ran 𝐹))
78 fo2nd 7141 . . . . . . . . . . . . . 14 2nd :V–onto→V
79 fofn 6079 . . . . . . . . . . . . . 14 (2nd :V–onto→V → 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6031 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd𝐹) Fn ℕ)
8280, 63, 81sylancr 694 . . . . . . . . . . . 12 (𝜑 → (2nd𝐹) Fn ℕ)
83 fnfun 5951 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → Fun (2nd𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd𝐹))
85 fndm 5953 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → dom (2nd𝐹) = ℕ)
86 eqimss2 3642 . . . . . . . . . . . 12 (dom (2nd𝐹) = ℕ → ℕ ⊆ dom (2nd𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (2nd𝐹))
88 dfimafn2 6208 . . . . . . . . . . 11 ((Fun (2nd𝐹) ∧ ℕ ⊆ dom (2nd𝐹)) → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
8984, 87, 88syl2anc 692 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
90 fnima 5972 . . . . . . . . . . 11 ((2nd𝐹) Fn ℕ → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9182, 90syl 17 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9289, 91eqtr3d 2657 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = ran (2nd𝐹))
93 rnco2 5606 . . . . . . . . 9 ran (2nd𝐹) = (2nd “ ran 𝐹)
9492, 93syl6eq 2671 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = (2nd “ ran 𝐹))
9577, 94uneq12d 3751 . . . . . . 7 (𝜑 → ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9657, 95syl5eq 2667 . . . . . 6 (𝜑 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9756, 96uneq12d 3751 . . . . 5 (𝜑 → ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9849, 97syl5eq 2667 . . . 4 (𝜑 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9937, 48, 983eqtr3d 2663 . . 3 (𝜑 ran ([,] ∘ 𝐹) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
1001, 99syl5sseqr 3638 . 2 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
101 ovolficcss 23161 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
1022, 101syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
103102ssdifssd 3731 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104 omelon 8495 . . . . . . . . . . 11 ω ∈ On
105 nnenom 12727 . . . . . . . . . . . 12 ℕ ≈ ω
106105ensymi 7958 . . . . . . . . . . 11 ω ≈ ℕ
107 isnumi 8724 . . . . . . . . . . 11 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108104, 106, 107mp2an 707 . . . . . . . . . 10 ℕ ∈ dom card
109 fofun 6078 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 3609 . . . . . . . . . . . . 13 ran 𝐹 ⊆ V
112 fof 6077 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
114113fdmi 6014 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtr4i 3622 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 1st
116 fores 6086 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹))
117110, 115, 116mp2an 707 . . . . . . . . . . 11 (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹)
118 ffn 6007 . . . . . . . . . . . . 13 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
1192, 118syl 17 . . . . . . . . . . . 12 (𝜑𝐹 Fn ℕ)
120 dffn4 6083 . . . . . . . . . . . 12 (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
121119, 120sylib 208 . . . . . . . . . . 11 (𝜑𝐹:ℕ–onto→ran 𝐹)
122 foco 6087 . . . . . . . . . . 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 694 . . . . . . . . . 10 (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
124 fodomnum 8832 . . . . . . . . . 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 7967 . . . . . . . . 9 (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
127125, 105, 126sylancl 693 . . . . . . . 8 (𝜑 → (1st “ ran 𝐹) ≼ ω)
128 fofun 6078 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
12978, 128ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
130 fof 6077 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
13178, 130ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
132131fdmi 6014 . . . . . . . . . . . . 13 dom 2nd = V
133111, 132sseqtr4i 3622 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 2nd
134 fores 6086 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹))
135129, 133, 134mp2an 707 . . . . . . . . . . 11 (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹)
136 foco 6087 . . . . . . . . . . 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 694 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
138 fodomnum 8832 . . . . . . . . . 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 7967 . . . . . . . . 9 (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
141139, 105, 140sylancl 693 . . . . . . . 8 (𝜑 → (2nd “ ran 𝐹) ≼ ω)
142 unctb 8979 . . . . . . . 8 (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143127, 141, 142syl2anc 692 . . . . . . 7 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
144 reldom 7913 . . . . . . . 8 Rel ≼
145144brrelexi 5123 . . . . . . 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 3608 . . . . . . . 8 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
148147, 99syl5sseq 3637 . . . . . . 7 (𝜑 ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
149 ssundif 4029 . . . . . . 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 7953 . . . . . 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 7961 . . . . 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 692 . . . 4 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
155 domentr 7967 . . . 4 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
156154, 106, 155sylancl 693 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
157 ovolctb2 23183 . . 3 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
158103, 156, 157syl2anc 692 . 2 (𝜑 → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
159100, 158jca 554 1 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  𝒫 cpw 4135  {csn 4153  {cpr 4155  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618   × cxp 5077  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  ccom 5083  Oncon0 5687  Fun wfun 5846   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852  (class class class)co 6610  ωcom 7019  1st c1st 7118  2nd c2nd 7119  cen 7904  cdom 7905  cardccrd 8713  cr 9887  0cc0 9888  *cxr 10025  cle 10027  cn 10972  (,)cioo 12125  [,]cicc 12128  vol*covol 23154
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 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
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-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-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-acn 8720  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-xadd 11899  df-ioo 12129  df-ico 12131  df-icc 12132  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-sum 14359  df-xmet 19671  df-met 19672  df-ovol 23156
This theorem is referenced by:  uniioombllem3  23276  uniioombllem4  23277  uniioombllem5  23278  uniiccmbl  23281
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