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Theorem uniiccdif 24189
 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 4099 . . 3 ran ((,) ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 24077 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
42, 3sylan 583 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
5 rexr 10678 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ∈ ℝ → (1st ‘(𝐹𝑥)) ∈ ℝ*)
6 rexr 10678 . . . . . . . 8 ((2nd ‘(𝐹𝑥)) ∈ ℝ → (2nd ‘(𝐹𝑥)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)) → (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)))
8 prunioo 12861 . . . . . . . 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 1157 . . . . . . 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 6737 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
122, 11sylan 583 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
132ffvelrnda 6828 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1413elin2d 4126 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
15 1st2nd2 7712 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1716fveq2d 6649 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
18 df-ov 7138 . . . . . . . . 9 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1917, 18eqtr4di 2851 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2012, 19eqtrd 2833 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
21 df-pr 4528 . . . . . . . 8 {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})
22 fvco3 6737 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
232, 22sylan 583 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
24 fvco3 6737 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
252, 24sylan 583 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
2623, 25preq12d 4637 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2721, 26syl5eqr 2847 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2820, 27uneq12d 4091 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}))
29 fvco3 6737 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
302, 29sylan 583 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3116fveq2d 6649 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
32 df-ov 7138 . . . . . . . 8 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3331, 32eqtr4di 2851 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3430, 33eqtrd 2833 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3510, 28, 343eqtr4rd 2844 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
3635iuneq2dv 4905 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
37 iccf 12828 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38 ffn 6487 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
3937, 38ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
40 inss2 4156 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41 rexpssxrxp 10677 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4240, 41sstri 3924 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43 fss 6501 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
442, 42, 43sylancl 589 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
45 fnfco 6517 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4639, 44, 45sylancr 590 . . . . 5 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47 fniunfv 6984 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
49 iunun 4978 . . . . 5 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}))
50 ioof 12827 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51 ffn 6487 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* × ℝ*)
53 fnfco 6517 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
5452, 44, 53sylancr 590 . . . . . . 7 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55 fniunfv 6984 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
57 iunun 4978 . . . . . . 7 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
58 fo1st 7693 . . . . . . . . . . . . . 14 1st :V–onto→V
59 fofn 6567 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 3939 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62 fss 6501 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
632, 61, 62sylancl 589 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶V)
64 fnfco 6517 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st𝐹) Fn ℕ)
6560, 63, 64sylancr 590 . . . . . . . . . . . 12 (𝜑 → (1st𝐹) Fn ℕ)
66 fnfun 6423 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → Fun (1st𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (𝜑 → Fun (1st𝐹))
68 fndm 6425 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → dom (1st𝐹) = ℕ)
69 eqimss2 3972 . . . . . . . . . . . 12 (dom (1st𝐹) = ℕ → ℕ ⊆ dom (1st𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (1st𝐹))
71 dfimafn2 6704 . . . . . . . . . . 11 ((Fun (1st𝐹) ∧ ℕ ⊆ dom (1st𝐹)) → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
7267, 70, 71syl2anc 587 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)})
73 fnima 6450 . . . . . . . . . . 11 ((1st𝐹) Fn ℕ → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7465, 73syl 17 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7572, 74eqtr3d 2835 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = ran (1st𝐹))
76 rnco2 6073 . . . . . . . . 9 ran (1st𝐹) = (1st “ ran 𝐹)
7775, 76eqtrdi 2849 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = (1st “ ran 𝐹))
78 fo2nd 7694 . . . . . . . . . . . . . 14 2nd :V–onto→V
79 fofn 6567 . . . . . . . . . . . . . 14 (2nd :V–onto→V → 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6517 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd𝐹) Fn ℕ)
8280, 63, 81sylancr 590 . . . . . . . . . . . 12 (𝜑 → (2nd𝐹) Fn ℕ)
83 fnfun 6423 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → Fun (2nd𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd𝐹))
85 fndm 6425 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → dom (2nd𝐹) = ℕ)
86 eqimss2 3972 . . . . . . . . . . . 12 (dom (2nd𝐹) = ℕ → ℕ ⊆ dom (2nd𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (2nd𝐹))
88 dfimafn2 6704 . . . . . . . . . . 11 ((Fun (2nd𝐹) ∧ ℕ ⊆ dom (2nd𝐹)) → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
8984, 87, 88syl2anc 587 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
90 fnima 6450 . . . . . . . . . . 11 ((2nd𝐹) Fn ℕ → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9182, 90syl 17 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9289, 91eqtr3d 2835 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = ran (2nd𝐹))
93 rnco2 6073 . . . . . . . . 9 ran (2nd𝐹) = (2nd “ ran 𝐹)
9492, 93eqtrdi 2849 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = (2nd “ ran 𝐹))
9577, 94uneq12d 4091 . . . . . . 7 (𝜑 → ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9657, 95syl5eq 2845 . . . . . 6 (𝜑 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9756, 96uneq12d 4091 . . . . 5 (𝜑 → ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9849, 97syl5eq 2845 . . . 4 (𝜑 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9936, 48, 983eqtr3d 2841 . . 3 (𝜑 ran ([,] ∘ 𝐹) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
1001, 99sseqtrrid 3968 . 2 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
101 ovolficcss 24080 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
1022, 101syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
103102ssdifssd 4070 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104 omelon 9095 . . . . . . . . . . 11 ω ∈ On
105 nnenom 13345 . . . . . . . . . . . 12 ℕ ≈ ω
106105ensymi 8544 . . . . . . . . . . 11 ω ≈ ℕ
107 isnumi 9361 . . . . . . . . . . 11 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108104, 106, 107mp2an 691 . . . . . . . . . 10 ℕ ∈ dom card
109 fofun 6566 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 3939 . . . . . . . . . . . . 13 ran 𝐹 ⊆ V
112 fof 6565 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
114113fdmi 6498 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtrri 3952 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 1st
116 fores 6575 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹))
117110, 115, 116mp2an 691 . . . . . . . . . . 11 (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹)
1182ffnd 6488 . . . . . . . . . . . 12 (𝜑𝐹 Fn ℕ)
119 dffn4 6571 . . . . . . . . . . . 12 (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120118, 119sylib 221 . . . . . . . . . . 11 (𝜑𝐹:ℕ–onto→ran 𝐹)
121 foco 6577 . . . . . . . . . . 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 590 . . . . . . . . . 10 (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123 fodomnum 9470 . . . . . . . . . 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 8553 . . . . . . . . 9 (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126124, 105, 125sylancl 589 . . . . . . . 8 (𝜑 → (1st “ ran 𝐹) ≼ ω)
127 fofun 6566 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
12878, 127ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
129 fof 6565 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
13078, 129ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
131130fdmi 6498 . . . . . . . . . . . . 13 dom 2nd = V
132111, 131sseqtrri 3952 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 2nd
133 fores 6575 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹))
134128, 132, 133mp2an 691 . . . . . . . . . . 11 (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹)
135 foco 6577 . . . . . . . . . . 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 590 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137 fodomnum 9470 . . . . . . . . . 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 8553 . . . . . . . . 9 (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140138, 105, 139sylancl 589 . . . . . . . 8 (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141 unctb 9618 . . . . . . . 8 (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
142126, 140, 141syl2anc 587 . . . . . . 7 (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143 ctex 8509 . . . . . . 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 3937 . . . . . . . 8 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
146145, 99sseqtrid 3967 . . . . . . 7 (𝜑 ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147 ssundif 4391 . . . . . . 7 ( ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))) ↔ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
148146, 147sylib 221 . . . . . 6 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149 ssdomg 8540 . . . . . 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 8547 . . . . 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 587 . . . 4 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω)
153 domentr 8553 . . . 4 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
154152, 106, 153sylancl 589 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ)
155 ovolctb2 24103 . . 3 ((( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
156103, 154, 155syl2anc 587 . 2 (𝜑 → (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0)
157100, 156jca 515 1 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  {cpr 4527  ⟨cop 4531  ∪ cuni 4800  ∪ ciun 4881   class class class wbr 5030   × cxp 5517  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522   ∘ ccom 5523  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135  ωcom 7562  1st c1st 7671  2nd c2nd 7672   ≈ cen 8491   ≼ cdom 8492  cardccrd 9350  ℝcr 10527  0cc0 10528  ℝ*cxr 10665   ≤ cle 10667  ℕcn 11627  (,)cioo 12728  [,]cicc 12731  vol*covol 24073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-inf 8893  df-oi 8960  df-dju 9316  df-card 9354  df-acn 9357  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-q 12339  df-rp 12380  df-xadd 12498  df-ioo 12732  df-ico 12734  df-icc 12735  df-fz 12888  df-fzo 13031  df-seq 13367  df-exp 13428  df-hash 13689  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-sum 15037  df-xmet 20087  df-met 20088  df-ovol 24075 This theorem is referenced by:  uniioombllem3  24196  uniioombllem4  24197  uniioombllem5  24198  uniiccmbl  24201
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