Theorem uniiccdif 24475

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 4086	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 24363	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
5		rexr 10879	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥)) ∈ ℝ → (1st ‘(𝐹‘𝑥)) ∈ ℝ*)
6		rexr 10879	. . . . . . . 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 13069	. . . . . . . 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 1162	. . . . . . 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 6810	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelrnda 6904	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4113	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 7800	. . . . . . . . . . 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 6721	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7216	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
21		df-pr 4544	. . . . . . . 8 ⊢ {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})
22		fvco3 6810	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
23	2, 22	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
24		fvco3 6810	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4657	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4078	. . . . . 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 6810	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7216	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2788	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 4928	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
37		iccf 13036	. . . . . . 7 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38		ffn 6545	. . . . . . 7 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ* × ℝ*)
40		inss2 4144	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 10878	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42	40, 41	sstri 3910	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43		fss 6562	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
44	2, 42, 43	sylancl 589	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
45		fnfco 6584	. . . . . 6 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 590	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7060	. . . . 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 5001	. . . . 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 13035	. . . . . . . . 9 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51		ffn 6545	. . . . . . . . 9 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ* × ℝ*)
53		fnfco 6584	. . . . . . . 8 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 590	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7060	. . . . . . 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 5001	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
58		fo1st 7781	. . . . . . . . . . . . . 14 ⊢ 1st :V–onto→V
59		fofn 6635	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st Fn V
61		ssv 3925	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6562	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 589	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6584	. . . . . . . . . . . . 13 ⊢ ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ∘ 𝐹) Fn ℕ)
66		fnfun 6479	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → Fun (1st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1st ∘ 𝐹))
68		fndm 6481	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → dom (1st ∘ 𝐹) = ℕ)
69		eqimss2 3958	. . . . . . . . . . . 12 ⊢ (dom (1st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1st ∘ 𝐹))
71		dfimafn2 6776	. . . . . . . . . . 11 ⊢ ((Fun (1st ∘ 𝐹) ∧ ℕ ⊆ dom (1st ∘ 𝐹)) → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
72	67, 70, 71	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
73		fnima 6508	. . . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = ran (1st ∘ 𝐹))
76		rnco2 6117	. . . . . . . . 9 ⊢ ran (1st ∘ 𝐹) = (1st “ ran 𝐹)
77	75, 76	eqtrdi 2794	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = (1st “ ran 𝐹))
78		fo2nd 7782	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
79		fofn 6635	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → 2nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2nd Fn V
81		fnfco 6584	. . . . . . . . . . . . 13 ⊢ ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ∘ 𝐹) Fn ℕ)
83		fnfun 6479	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → Fun (2nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2nd ∘ 𝐹))
85		fndm 6481	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → dom (2nd ∘ 𝐹) = ℕ)
86		eqimss2 3958	. . . . . . . . . . . 12 ⊢ (dom (2nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2nd ∘ 𝐹))
88		dfimafn2 6776	. . . . . . . . . . 11 ⊢ ((Fun (2nd ∘ 𝐹) ∧ ℕ ⊆ dom (2nd ∘ 𝐹)) → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
89	84, 87, 88	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
90		fnima 6508	. . . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = ran (2nd ∘ 𝐹))
93		rnco2 6117	. . . . . . . . 9 ⊢ ran (2nd ∘ 𝐹) = (2nd “ ran 𝐹)
94	92, 93	eqtrdi 2794	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = (2nd “ ran 𝐹))
95	77, 94	uneq12d 4078	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
96	57, 95	syl5eq 2790	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
97	56, 96	uneq12d 4078	. . . . 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 2790	. . . 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 2785	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
100	1, 99	sseqtrrid 3954	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 24366	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4057	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9261	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13553	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 8678	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9562	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 692	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6634	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
111		ssv 3925	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6633	. . . . . . . . . . . . . . 15 ⊢ (1st :V–onto→V → 1st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1st :V⟶V
114	113	fdmi 6557	. . . . . . . . . . . . 13 ⊢ dom 1st = V
115	111, 114	sseqtrri 3938	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1st
116		fores 6643	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ ran 𝐹 ⊆ dom 1st ) → (1st ↾ ran 𝐹):ran 𝐹–onto→(1st “ ran 𝐹))
117	110, 115, 116	mp2an 692	. . . . . . . . . . 11 ⊢ (1st ↾ ran 𝐹):ran 𝐹–onto→(1st “ ran 𝐹)
118	2	ffnd 6546	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6639	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6647	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123		fodomnum 9671	. . . . . . . . . 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 8687	. . . . . . . . 9 ⊢ (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 589	. . . . . . . 8 ⊢ (𝜑 → (1st “ ran 𝐹) ≼ ω)
127		fofun 6634	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
129		fof 6633	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
131	130	fdmi 6557	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
132	111, 131	sseqtrri 3938	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2nd
133		fores 6643	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ ran 𝐹 ⊆ dom 2nd ) → (2nd ↾ ran 𝐹):ran 𝐹–onto→(2nd “ ran 𝐹))
134	128, 132, 133	mp2an 692	. . . . . . . . . . 11 ⊢ (2nd ↾ ran 𝐹):ran 𝐹–onto→(2nd “ ran 𝐹)
135		foco 6647	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137		fodomnum 9671	. . . . . . . . . 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 8687	. . . . . . . . 9 ⊢ (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 589	. . . . . . . 8 ⊢ (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141		unctb 9819	. . . . . . . 8 ⊢ (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
142	126, 140, 141	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143		ctex 8643	. . . . . . 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 3923	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 3953	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147		ssundif 4399	. . . . . . 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 221	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149		ssdomg 8674	. . . . . 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 8681	. . . . 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 587	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 8687	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 589	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 24389	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 587	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 515	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  𝒫 cpw 4513  {csn 4541  {cpr 4543  ⟨cop 4547  ∪ cuni 4819  ∪ ciun 4904   class class class wbr 5053   × cxp 5549  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554   ∘ ccom 5555  Oncon0 6213  Fun wfun 6374   Fn wfn 6375  ⟶wf 6376  –onto→wfo 6378  ‘cfv 6380  (class class class)co 7213  ωcom 7644  1^st c1st 7759  2^nd c2nd 7760   ≈ cen 8623   ≼ cdom 8624  cardccrd 9551  ℝcr 10728  0cc0 10729  ℝ^*cxr 10866   ≤ cle 10868  ℕcn 11830  (,)cioo 12935  [,]cicc 12938  vol*covol 24359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-acn 9558  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xadd 12705  df-ioo 12939  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250  df-xmet 20356  df-met 20357  df-ovol 24361

This theorem is referenced by:  uniioombllem3  24482  uniioombllem4  24483  uniioombllem5  24484  uniiccmbl  24487