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Theorem uniiccdif 25613
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 4178 . . 3 ran ((,) ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 25501 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
42, 3sylan 580 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
5 rexr 11307 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ∈ ℝ → (1st ‘(𝐹𝑥)) ∈ ℝ*)
6 rexr 11307 . . . . . . . 8 ((2nd ‘(𝐹𝑥)) ∈ ℝ → (2nd ‘(𝐹𝑥)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)) → (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)))
8 prunioo 13521 . . . . . . . 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 1161 . . . . . . 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 7008 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
122, 11sylan 580 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
132ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1413elin2d 4205 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
15 1st2nd2 8053 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1716fveq2d 6910 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
18 df-ov 7434 . . . . . . . . 9 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1917, 18eqtr4di 2795 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2012, 19eqtrd 2777 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
21 df-pr 4629 . . . . . . . 8 {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})
22 fvco3 7008 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
232, 22sylan 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
24 fvco3 7008 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
252, 24sylan 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
2623, 25preq12d 4741 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2721, 26eqtr3id 2791 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2820, 27uneq12d 4169 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}))
29 fvco3 7008 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
302, 29sylan 580 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3116fveq2d 6910 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
32 df-ov 7434 . . . . . . . 8 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3331, 32eqtr4di 2795 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3430, 33eqtrd 2777 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3510, 28, 343eqtr4rd 2788 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
3635iuneq2dv 5016 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
37 iccf 13488 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38 ffn 6736 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
3937, 38ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
40 inss2 4238 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41 rexpssxrxp 11306 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4240, 41sstri 3993 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43 fss 6752 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
442, 42, 43sylancl 586 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
45 fnfco 6773 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4639, 44, 45sylancr 587 . . . . 5 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47 fniunfv 7267 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
49 iunun 5093 . . . . 5 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}))
50 ioof 13487 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51 ffn 6736 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* × ℝ*)
53 fnfco 6773 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
5452, 44, 53sylancr 587 . . . . . . 7 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55 fniunfv 7267 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
57 iunun 5093 . . . . . . 7 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
58 fo1st 8034 . . . . . . . . . . . . . 14 1st :V–onto→V
59 fofn 6822 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 4008 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62 fss 6752 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
632, 61, 62sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶V)
64 fnfco 6773 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st𝐹) Fn ℕ)
6560, 63, 64sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐹) Fn ℕ)
66 fnfun 6668 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → Fun (1st𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (𝜑 → Fun (1st𝐹))
68 fndm 6671 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → dom (1st𝐹) = ℕ)
69 eqimss2 4043 . . . . . . . . . . . 12 (dom (1st𝐹) = ℕ → ℕ ⊆ dom (1st𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (1st𝐹))
71 dfimafn2 6972 . . . . . . . . . . 11 ((Fun (1st𝐹) ∧ ℕ ⊆ dom (1st𝐹)) → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
7267, 70, 71syl2anc 584 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
73 fnima 6698 . . . . . . . . . . 11 ((1st𝐹) Fn ℕ → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7465, 73syl 17 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7572, 74eqtr3d 2779 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = ran (1st𝐹))
76 rnco2 6273 . . . . . . . . 9 ran (1st𝐹) = (1st “ ran 𝐹)
7775, 76eqtrdi 2793 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = (1st “ ran 𝐹))
78 fo2nd 8035 . . . . . . . . . . . . . 14 2nd :V–onto→V
79 fofn 6822 . . . . . . . . . . . . . 14 (2nd :V–onto→V → 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6773 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd𝐹) Fn ℕ)
8280, 63, 81sylancr 587 . . . . . . . . . . . 12 (𝜑 → (2nd𝐹) Fn ℕ)
83 fnfun 6668 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → Fun (2nd𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd𝐹))
85 fndm 6671 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → dom (2nd𝐹) = ℕ)
86 eqimss2 4043 . . . . . . . . . . . 12 (dom (2nd𝐹) = ℕ → ℕ ⊆ dom (2nd𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (2nd𝐹))
88 dfimafn2 6972 . . . . . . . . . . 11 ((Fun (2nd𝐹) ∧ ℕ ⊆ dom (2nd𝐹)) → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
8984, 87, 88syl2anc 584 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
90 fnima 6698 . . . . . . . . . . 11 ((2nd𝐹) Fn ℕ → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9182, 90syl 17 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9289, 91eqtr3d 2779 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = ran (2nd𝐹))
93 rnco2 6273 . . . . . . . . 9 ran (2nd𝐹) = (2nd “ ran 𝐹)
9492, 93eqtrdi 2793 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = (2nd “ ran 𝐹))
9577, 94uneq12d 4169 . . . . . . 7 (𝜑 → ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9657, 95eqtrid 2789 . . . . . 6 (𝜑 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9756, 96uneq12d 4169 . . . . 5 (𝜑 → ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9849, 97eqtrid 2789 . . . 4 (𝜑 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9936, 48, 983eqtr3d 2785 . . 3 (𝜑 ran ([,] ∘ 𝐹) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
1001, 99sseqtrrid 4027 . 2 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
101 ovolficcss 25504 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
1022, 101syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
103102ssdifssd 4147 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104 omelon 9686 . . . . . . . . . . 11 ω ∈ On
105 nnenom 14021 . . . . . . . . . . . 12 ℕ ≈ ω
106105ensymi 9044 . . . . . . . . . . 11 ω ≈ ℕ
107 isnumi 9986 . . . . . . . . . . 11 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108104, 106, 107mp2an 692 . . . . . . . . . 10 ℕ ∈ dom card
109 fofun 6821 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 4008 . . . . . . . . . . . . 13 ran 𝐹 ⊆ V
112 fof 6820 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
114113fdmi 6747 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtrri 4033 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 1st
116 fores 6830 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹))
117110, 115, 116mp2an 692 . . . . . . . . . . 11 (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹)
1182ffnd 6737 . . . . . . . . . . . 12 (𝜑𝐹 Fn ℕ)
119 dffn4 6826 . . . . . . . . . . . 12 (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120118, 119sylib 218 . . . . . . . . . . 11 (𝜑𝐹:ℕ–onto→ran 𝐹)
121 foco 6834 . . . . . . . . . . 11 (((1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
122117, 120, 121sylancr 587 . . . . . . . . . 10 (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123 fodomnum 10097 . . . . . . . . . 10 (ℕ ∈ dom card → (((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹) → (1st “ ran 𝐹) ≼ ℕ))
124108, 122, 123mpsyl 68 . . . . . . . . 9 (𝜑 → (1st “ ran 𝐹) ≼ ℕ)
125 domentr 9053 . . . . . . . . 9 (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126124, 105, 125sylancl 586 . . . . . . . 8 (𝜑 → (1st “ ran 𝐹) ≼ ω)
127 fofun 6821 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
12878, 127ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
129 fof 6820 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
13078, 129ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
131130fdmi 6747 . . . . . . . . . . . . 13 dom 2nd = V
132111, 131sseqtrri 4033 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 2nd
133 fores 6830 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹))
134128, 132, 133mp2an 692 . . . . . . . . . . 11 (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹)
135 foco 6834 . . . . . . . . . . 11 (((2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
136134, 120, 135sylancr 587 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137 fodomnum 10097 . . . . . . . . . 10 (ℕ ∈ dom card → (((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹) → (2nd “ ran 𝐹) ≼ ℕ))
138108, 136, 137mpsyl 68 . . . . . . . . 9 (𝜑 → (2nd “ ran 𝐹) ≼ ℕ)
139 domentr 9053 . . . . . . . . 9 (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140138, 105, 139sylancl 586 . . . . . . . 8 (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141 unctb 10244 . . . . . . . 8 (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
142126, 140, 141syl2anc 584 . . . . . . 7 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143 ctex 9004 . . . . . . 7 (((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
144142, 143syl 17 . . . . . 6 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
145 ssid 4006 . . . . . . . 8 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
146145, 99sseqtrid 4026 . . . . . . 7 (𝜑 ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147 ssundif 4488 . . . . . . 7 ( ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))) ↔ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
148146, 147sylib 218 . . . . . 6 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149 ssdomg 9040 . . . . . 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))))
150144, 148, 149sylc 65 . . . . 5 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
151 domtr 9047 . . . . 5 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∧ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
152150, 142, 151syl2anc 584 . . . 4 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
153 domentr 9053 . . . 4 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
154152, 106, 153sylancl 586 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
155 ovolctb2 25527 . . 3 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
156103, 154, 155syl2anc 584 . 2 (𝜑 → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
157100, 156jca 511 1 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626  {cpr 4628  cop 4632   cuni 4907   ciun 4991   class class class wbr 5143   × cxp 5683  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  2nd c2nd 8013  cen 8982  cdom 8983  cardccrd 9975  cr 11154  0cc0 11155  *cxr 11294  cle 11296  cn 12266  (,)cioo 13387  [,]cicc 13390  vol*covol 25497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499
This theorem is referenced by:  uniioombllem3  25620  uniioombllem4  25621  uniioombllem5  25622  uniiccmbl  25625
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