Theorem uniiccdif 24178

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	⊢ (𝜑 → 𝐹: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.

Step	Hyp	Ref	Expression
1		ssun1 4147	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 24066	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
4	2, 3	sylan 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
5		rexr 10686	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥)) ∈ ℝ → (1st ‘(𝐹‘𝑥)) ∈ ℝ*)
6		rexr 10686	. . . . . . . 8 ⊢ ((2nd ‘(𝐹‘𝑥)) ∈ ℝ → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*)
7		id 22	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥)) → (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥)))
8		prunioo 12866	. . . . . . . 8 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ* ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ* ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))) → (((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∪ {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))}) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
9	5, 6, 7, 8	syl3an 1156	. . . . . . 7 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))) → (((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∪ {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))}) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
10	4, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∪ {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))}) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
11		fvco3 6759	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelrnda 6850	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4175	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 7727	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
17	16	fveq2d 6673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7158	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
19	17, 18	syl6eqr 2874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
21		df-pr 4569	. . . . . . . 8 ⊢ {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})
22		fvco3 6759	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
23	2, 22	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
24		fvco3 6759	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4676	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
27	21, 26	syl5eqr 2870	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})) = (((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∪ {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))}))
29		fvco3 6759	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6673	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7158	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
33	31, 32	syl6eqr 2874	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2856	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2867	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 4942	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
37		iccf 12835	. . . . . . 7 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38		ffn 6513	. . . . . . 7 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ* × ℝ*)
40		inss2 4205	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 10685	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42	40, 41	sstri 3975	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43		fss 6526	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
44	2, 42, 43	sylancl 588	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
45		fnfco 6542	. . . . . 6 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 589	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7005	. . . . 5 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
49		iunun 5014	. . . . 5 ⊢ ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})) = (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}))
50		ioof 12834	. . . . . . . . 9 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51		ffn 6513	. . . . . . . . 9 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ* × ℝ*)
53		fnfco 6542	. . . . . . . 8 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 589	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7005	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
56	54, 55	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
57		iunun 5014	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
58		fo1st 7708	. . . . . . . . . . . . . 14 ⊢ 1st :V–onto→V
59		fofn 6591	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st Fn V
61		ssv 3990	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6526	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 588	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6542	. . . . . . . . . . . . 13 ⊢ ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ∘ 𝐹) Fn ℕ)
66		fnfun 6452	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → Fun (1st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1st ∘ 𝐹))
68		fndm 6454	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → dom (1st ∘ 𝐹) = ℕ)
69		eqimss2 4023	. . . . . . . . . . . 12 ⊢ (dom (1st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1st ∘ 𝐹))
71		dfimafn2 6728	. . . . . . . . . . 11 ⊢ ((Fun (1st ∘ 𝐹) ∧ ℕ ⊆ dom (1st ∘ 𝐹)) → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
72	67, 70, 71	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
73		fnima 6477	. . . . . . . . . . 11 ⊢ ((1st ∘ 𝐹) Fn ℕ → ((1st ∘ 𝐹) “ ℕ) = ran (1st ∘ 𝐹))
74	65, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ∘ 𝐹) “ ℕ) = ran (1st ∘ 𝐹))
75	72, 74	eqtr3d 2858	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = ran (1st ∘ 𝐹))
76		rnco2 6105	. . . . . . . . 9 ⊢ ran (1st ∘ 𝐹) = (1st “ ran 𝐹)
77	75, 76	syl6eq 2872	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = (1st “ ran 𝐹))
78		fo2nd 7709	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
79		fofn 6591	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → 2nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2nd Fn V
81		fnfco 6542	. . . . . . . . . . . . 13 ⊢ ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ∘ 𝐹) Fn ℕ)
83		fnfun 6452	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → Fun (2nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2nd ∘ 𝐹))
85		fndm 6454	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → dom (2nd ∘ 𝐹) = ℕ)
86		eqimss2 4023	. . . . . . . . . . . 12 ⊢ (dom (2nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2nd ∘ 𝐹))
88		dfimafn2 6728	. . . . . . . . . . 11 ⊢ ((Fun (2nd ∘ 𝐹) ∧ ℕ ⊆ dom (2nd ∘ 𝐹)) → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
89	84, 87, 88	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
90		fnima 6477	. . . . . . . . . . 11 ⊢ ((2nd ∘ 𝐹) Fn ℕ → ((2nd ∘ 𝐹) “ ℕ) = ran (2nd ∘ 𝐹))
91	82, 90	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ∘ 𝐹) “ ℕ) = ran (2nd ∘ 𝐹))
92	89, 91	eqtr3d 2858	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = ran (2nd ∘ 𝐹))
93		rnco2 6105	. . . . . . . . 9 ⊢ ran (2nd ∘ 𝐹) = (2nd “ ran 𝐹)
94	92, 93	syl6eq 2872	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = (2nd “ ran 𝐹))
95	77, 94	uneq12d 4139	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
96	57, 95	syl5eq 2868	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
97	56, 96	uneq12d 4139	. . . . 5 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
98	49, 97	syl5eq 2868	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
99	36, 48, 98	3eqtr3d 2864	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
100	1, 99	sseqtrrid 4019	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 24069	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4118	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9108	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13347	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 8558	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9374	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 690	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6590	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
111		ssv 3990	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6589	. . . . . . . . . . . . . . 15 ⊢ (1st :V–onto→V → 1st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1st :V⟶V
114	113	fdmi 6523	. . . . . . . . . . . . 13 ⊢ dom 1st = V
115	111, 114	sseqtrri 4003	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1st
116		fores 6599	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹–onto→(1st “ ran 𝐹))
117	110, 115, 116	mp2an 690	. . . . . . . . . . 11 ⊢ (1st ↾ ran 𝐹):ran 𝐹–onto→(1st “ ran 𝐹)
118	2	ffnd 6514	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6595	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 220	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6601	. . . . . . . . . . 11 ⊢ (((1st ↾ ran 𝐹):ran 𝐹–onto→(1st “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
122	117, 120, 121	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123		fodomnum 9482	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹) → (1st “ ran 𝐹) ≼ ℕ))
124	108, 122, 123	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (1st “ ran 𝐹) ≼ ℕ)
125		domentr 8567	. . . . . . . . 9 ⊢ (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → (1st “ ran 𝐹) ≼ ω)
127		fofun 6590	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
129		fof 6589	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
131	130	fdmi 6523	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
132	111, 131	sseqtrri 4003	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2nd
133		fores 6599	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹–onto→(2nd “ ran 𝐹))
134	128, 132, 133	mp2an 690	. . . . . . . . . . 11 ⊢ (2nd ↾ ran 𝐹):ran 𝐹–onto→(2nd “ ran 𝐹)
135		foco 6601	. . . . . . . . . . 11 ⊢ (((2nd ↾ ran 𝐹):ran 𝐹–onto→(2nd “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
136	134, 120, 135	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137		fodomnum 9482	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹) → (2nd “ ran 𝐹) ≼ ℕ))
138	108, 136, 137	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (2nd “ ran 𝐹) ≼ ℕ)
139		domentr 8567	. . . . . . . . 9 ⊢ (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141		unctb 9626	. . . . . . . 8 ⊢ (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
142	126, 140, 141	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143		ctex 8523	. . . . . . 7 ⊢ (((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
144	142, 143	syl 17	. . . . . 6 ⊢ (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∈ V)
145		ssid 3988	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 4018	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147		ssundif 4432	. . . . . . 7 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))) ↔ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
148	146, 147	sylib 220	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149		ssdomg 8554	. . . . . 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))))
150	144, 148, 149	sylc 65	. . . . 5 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
151		domtr 8561	. . . . 5 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ∧ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
152	150, 142, 151	syl2anc 586	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 8567	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 588	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 24092	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 586	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 514	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  {csn 4566  {cpr 4568  ⟨cop 4572  ∪ cuni 4837  ∪ ciun 4918   class class class wbr 5065   × cxp 5552  dom cdm 5554  ran crn 5555   ↾ cres 5556   “ cima 5557   ∘ ccom 5558  Oncon0 6190  Fun wfun 6348   Fn wfn 6349  ⟶wf 6350  –onto→wfo 6352  ‘cfv 6354  (class class class)co 7155  ωcom 7579  1^st c1st 7686  2^nd c2nd 7687   ≈ cen 8505   ≼ cdom 8506  cardccrd 9363  ℝcr 10535  0cc0 10536  ℝ^*cxr 10673   ≤ cle 10675  ℕcn 11637  (,)cioo 12737  [,]cicc 12740  vol*covol 24062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-acn 9370  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-xadd 12507  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-sum 15042  df-xmet 20537  df-met 20538  df-ovol 24064

This theorem is referenced by:  uniioombllem3  24185  uniioombllem4  24186  uniioombllem5  24187  uniiccmbl  24190