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Theorem uniiccdif 24942
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 4132 . . 3 ran ((,) ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 24830 . . . . . . . 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 11201 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ∈ ℝ → (1st ‘(𝐹𝑥)) ∈ ℝ*)
6 rexr 11201 . . . . . . . 8 ((2nd ‘(𝐹𝑥)) ∈ ℝ → (2nd ‘(𝐹𝑥)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)) → (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥)))
8 prunioo 13398 . . . . . . . 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 1160 . . . . . . 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 6940 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
122, 11sylan 580 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
132ffvelcdmda 7035 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1413elin2d 4159 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
15 1st2nd2 7960 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1716fveq2d 6846 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
18 df-ov 7360 . . . . . . . . 9 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1917, 18eqtr4di 2794 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2012, 19eqtrd 2776 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
21 df-pr 4589 . . . . . . . 8 {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})
22 fvco3 6940 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
232, 22sylan 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
24 fvco3 6940 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
252, 24sylan 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
2623, 25preq12d 4702 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → {((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)} = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2721, 26eqtr3id 2790 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))})
2820, 27uneq12d 4124 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = (((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∪ {(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))}))
29 fvco3 6940 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
302, 29sylan 580 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3116fveq2d 6846 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
32 df-ov 7360 . . . . . . . 8 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3331, 32eqtr4di 2794 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3430, 33eqtrd 2776 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))))
3510, 28, 343eqtr4rd 2787 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
3635iuneq2dv 4978 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})))
37 iccf 13365 . . . . . . 7 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38 ffn 6668 . . . . . . 7 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
3937, 38ax-mp 5 . . . . . 6 [,] Fn (ℝ* × ℝ*)
40 inss2 4189 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41 rexpssxrxp 11200 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4240, 41sstri 3953 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43 fss 6685 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
442, 42, 43sylancl 586 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
45 fnfco 6707 . . . . . 6 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4639, 44, 45sylancr 587 . . . . 5 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47 fniunfv 7194 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
49 iunun 5053 . . . . 5 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}))
50 ioof 13364 . . . . . . . . 9 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51 ffn 6668 . . . . . . . . 9 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* × ℝ*)
53 fnfco 6707 . . . . . . . 8 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
5452, 44, 53sylancr 587 . . . . . . 7 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55 fniunfv 7194 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
57 iunun 5053 . . . . . . 7 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)})
58 fo1st 7941 . . . . . . . . . . . . . 14 1st :V–onto→V
59 fofn 6758 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 3968 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62 fss 6685 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
632, 61, 62sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶V)
64 fnfco 6707 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st𝐹) Fn ℕ)
6560, 63, 64sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐹) Fn ℕ)
66 fnfun 6602 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → Fun (1st𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (𝜑 → Fun (1st𝐹))
68 fndm 6605 . . . . . . . . . . . 12 ((1st𝐹) Fn ℕ → dom (1st𝐹) = ℕ)
69 eqimss2 4001 . . . . . . . . . . . 12 (dom (1st𝐹) = ℕ → ℕ ⊆ dom (1st𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (1st𝐹))
71 dfimafn2 6906 . . . . . . . . . . 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 6631 . . . . . . . . . . 11 ((1st𝐹) Fn ℕ → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7465, 73syl 17 . . . . . . . . . 10 (𝜑 → ((1st𝐹) “ ℕ) = ran (1st𝐹))
7572, 74eqtr3d 2778 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = ran (1st𝐹))
76 rnco2 6205 . . . . . . . . 9 ran (1st𝐹) = (1st “ ran 𝐹)
7775, 76eqtrdi 2792 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} = (1st “ ran 𝐹))
78 fo2nd 7942 . . . . . . . . . . . . . 14 2nd :V–onto→V
79 fofn 6758 . . . . . . . . . . . . . 14 (2nd :V–onto→V → 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6707 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd𝐹) Fn ℕ)
8280, 63, 81sylancr 587 . . . . . . . . . . . 12 (𝜑 → (2nd𝐹) Fn ℕ)
83 fnfun 6602 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → Fun (2nd𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd𝐹))
85 fndm 6605 . . . . . . . . . . . 12 ((2nd𝐹) Fn ℕ → dom (2nd𝐹) = ℕ)
86 eqimss2 4001 . . . . . . . . . . . 12 (dom (2nd𝐹) = ℕ → ℕ ⊆ dom (2nd𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (𝜑 → ℕ ⊆ dom (2nd𝐹))
88 dfimafn2 6906 . . . . . . . . . . 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 6631 . . . . . . . . . . 11 ((2nd𝐹) Fn ℕ → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9182, 90syl 17 . . . . . . . . . 10 (𝜑 → ((2nd𝐹) “ ℕ) = ran (2nd𝐹))
9289, 91eqtr3d 2778 . . . . . . . . 9 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = ran (2nd𝐹))
93 rnco2 6205 . . . . . . . . 9 ran (2nd𝐹) = (2nd “ ran 𝐹)
9492, 93eqtrdi 2792 . . . . . . . 8 (𝜑 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)} = (2nd “ ran 𝐹))
9577, 94uneq12d 4124 . . . . . . 7 (𝜑 → ( 𝑥 ∈ ℕ {((1st𝐹)‘𝑥)} ∪ 𝑥 ∈ ℕ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9657, 95eqtrid 2788 . . . . . 6 (𝜑 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
9756, 96uneq12d 4124 . . . . 5 (𝜑 → ( 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ 𝑥 ∈ ℕ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9849, 97eqtrid 2788 . . . 4 (𝜑 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st𝐹)‘𝑥)} ∪ {((2nd𝐹)‘𝑥)})) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
9936, 48, 983eqtr3d 2784 . . 3 (𝜑 ran ([,] ∘ 𝐹) = ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
1001, 99sseqtrrid 3997 . 2 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
101 ovolficcss 24833 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
1022, 101syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
103102ssdifssd 4102 . . 3 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104 omelon 9582 . . . . . . . . . . 11 ω ∈ On
105 nnenom 13885 . . . . . . . . . . . 12 ℕ ≈ ω
106105ensymi 8944 . . . . . . . . . . 11 ω ≈ ℕ
107 isnumi 9882 . . . . . . . . . . 11 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108104, 106, 107mp2an 690 . . . . . . . . . 10 ℕ ∈ dom card
109 fofun 6757 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 3968 . . . . . . . . . . . . 13 ran 𝐹 ⊆ V
112 fof 6756 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
114113fdmi 6680 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtrri 3981 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 1st
116 fores 6766 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹))
117110, 115, 116mp2an 690 . . . . . . . . . . 11 (1st ↾ ran 𝐹):ran 𝐹onto→(1st “ ran 𝐹)
1182ffnd 6669 . . . . . . . . . . . 12 (𝜑𝐹 Fn ℕ)
119 dffn4 6762 . . . . . . . . . . . 12 (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120118, 119sylib 217 . . . . . . . . . . 11 (𝜑𝐹:ℕ–onto→ran 𝐹)
121 foco 6770 . . . . . . . . . . 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 9993 . . . . . . . . . 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 8953 . . . . . . . . 9 (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126124, 105, 125sylancl 586 . . . . . . . 8 (𝜑 → (1st “ ran 𝐹) ≼ ω)
127 fofun 6757 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
12878, 127ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
129 fof 6756 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
13078, 129ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
131130fdmi 6680 . . . . . . . . . . . . 13 dom 2nd = V
132111, 131sseqtrri 3981 . . . . . . . . . . . 12 ran 𝐹 ⊆ dom 2nd
133 fores 6766 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹))
134128, 132, 133mp2an 690 . . . . . . . . . . 11 (2nd ↾ ran 𝐹):ran 𝐹onto→(2nd “ ran 𝐹)
135 foco 6770 . . . . . . . . . . 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 9993 . . . . . . . . . 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 8953 . . . . . . . . 9 (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140138, 105, 139sylancl 586 . . . . . . . 8 (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141 unctb 10141 . . . . . . . 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 8903 . . . . . . 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 3966 . . . . . . . 8 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
146145, 99sseqtrid 3996 . . . . . . 7 (𝜑 ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147 ssundif 4445 . . . . . . 7 ( ran ([,] ∘ 𝐹) ⊆ ( ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))) ↔ ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
148146, 147sylib 217 . . . . . 6 (𝜑 → ( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149 ssdomg 8940 . . . . . 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 8947 . . . . 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 8953 . . . 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 24856 . . 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 512 1 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586  {cpr 4588  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Oncon0 6317  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  ωcom 7802  1st c1st 7919  2nd c2nd 7920  cen 8880  cdom 8881  cardccrd 9871  cr 11050  0cc0 11051  *cxr 11188  cle 11190  cn 12153  (,)cioo 13264  [,]cicc 13267  vol*covol 24826
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828
This theorem is referenced by:  uniioombllem3  24949  uniioombllem4  24950  uniioombllem5  24951  uniiccmbl  24954
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