Theorem uniiccdif 25507

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 4127	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 25395	. . . . . . . 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 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
5		rexr 11165	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥)) ∈ ℝ → (1st ‘(𝐹‘𝑥)) ∈ ℝ*)
6		rexr 11165	. . . . . . . 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 13383	. . . . . . . 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 1160	. . . . . . 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 6927	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelcdmda 7023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 7966	. . . . . . . . . . 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 6832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7355	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
21		df-pr 4578	. . . . . . . 8 ⊢ {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})
22		fvco3 6927	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
23	2, 22	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
24		fvco3 6927	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4118	. . . . . 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 6927	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7355	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 4966	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
37		iccf 13350	. . . . . . 7 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38		ffn 6656	. . . . . . 7 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ* × ℝ*)
40		inss2 4187	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 11164	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42	40, 41	sstri 3940	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43		fss 6672	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
44	2, 42, 43	sylancl 586	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
45		fnfco 6693	. . . . . 6 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 587	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7187	. . . . 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 5043	. . . . 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 13349	. . . . . . . . 9 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51		ffn 6656	. . . . . . . . 9 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ* × ℝ*)
53		fnfco 6693	. . . . . . . 8 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 587	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7187	. . . . . . 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 5043	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
58		fo1st 7947	. . . . . . . . . . . . . 14 ⊢ 1st :V–onto→V
59		fofn 6742	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st Fn V
61		ssv 3955	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6672	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6693	. . . . . . . . . . . . 13 ⊢ ((1st Fn V ∧ 𝐹:ℕ⟶V) → (1st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ∘ 𝐹) Fn ℕ)
66		fnfun 6586	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → Fun (1st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1st ∘ 𝐹))
68		fndm 6589	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → dom (1st ∘ 𝐹) = ℕ)
69		eqimss2 3990	. . . . . . . . . . . 12 ⊢ (dom (1st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1st ∘ 𝐹))
71		dfimafn2 6891	. . . . . . . . . . 11 ⊢ ((Fun (1st ∘ 𝐹) ∧ ℕ ⊆ dom (1st ∘ 𝐹)) → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
72	67, 70, 71	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)})
73		fnima 6616	. . . . . . . . . . 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 2770	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = ran (1st ∘ 𝐹))
76		rnco2 6206	. . . . . . . . 9 ⊢ ran (1st ∘ 𝐹) = (1st “ ran 𝐹)
77	75, 76	eqtrdi 2784	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = (1st “ ran 𝐹))
78		fo2nd 7948	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
79		fofn 6742	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → 2nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2nd Fn V
81		fnfco 6693	. . . . . . . . . . . . 13 ⊢ ((2nd Fn V ∧ 𝐹:ℕ⟶V) → (2nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ∘ 𝐹) Fn ℕ)
83		fnfun 6586	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → Fun (2nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2nd ∘ 𝐹))
85		fndm 6589	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → dom (2nd ∘ 𝐹) = ℕ)
86		eqimss2 3990	. . . . . . . . . . . 12 ⊢ (dom (2nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2nd ∘ 𝐹))
88		dfimafn2 6891	. . . . . . . . . . 11 ⊢ ((Fun (2nd ∘ 𝐹) ∧ ℕ ⊆ dom (2nd ∘ 𝐹)) → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
89	84, 87, 88	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
90		fnima 6616	. . . . . . . . . . 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 2770	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = ran (2nd ∘ 𝐹))
93		rnco2 6206	. . . . . . . . 9 ⊢ ran (2nd ∘ 𝐹) = (2nd “ ran 𝐹)
94	92, 93	eqtrdi 2784	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = (2nd “ ran 𝐹))
95	77, 94	uneq12d 4118	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
96	57, 95	eqtrid 2780	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
97	56, 96	uneq12d 4118	. . . . 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	eqtrid 2780	. . . 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 2776	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
100	1, 99	sseqtrrid 3974	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 25398	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4096	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9543	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13889	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 8933	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9846	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 692	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6741	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
111		ssv 3955	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6740	. . . . . . . . . . . . . . 15 ⊢ (1st :V–onto→V → 1st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1st :V⟶V
114	113	fdmi 6667	. . . . . . . . . . . . 13 ⊢ dom 1st = V
115	111, 114	sseqtrri 3980	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1st
116		fores 6750	. . . . . . . . . . . 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 6657	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6746	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6754	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1st “ ran 𝐹))
123		fodomnum 9955	. . . . . . . . . 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 8942	. . . . . . . . 9 ⊢ (((1st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (1st “ ran 𝐹) ≼ ω)
127		fofun 6741	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
129		fof 6740	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
131	130	fdmi 6667	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
132	111, 131	sseqtrri 3980	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2nd
133		fores 6750	. . . . . . . . . . . 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 6754	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2nd “ ran 𝐹))
137		fodomnum 9955	. . . . . . . . . 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 8942	. . . . . . . . 9 ⊢ (((2nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (2nd “ ran 𝐹) ≼ ω)
141		unctb 10102	. . . . . . . 8 ⊢ (((1st “ ran 𝐹) ≼ ω ∧ (2nd “ ran 𝐹) ≼ ω) → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
142	126, 140, 141	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)) ≼ ω)
143		ctex 8892	. . . . . . 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 3953	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 3973	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147		ssundif 4437	. . . . . . 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 218	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
149		ssdomg 8929	. . . . . 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 8936	. . . . 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 584	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 8942	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 586	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 25421	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 584	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 511	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549  {csn 4575  {cpr 4577  ⟨cop 4581  ∪ cuni 4858  ∪ ciun 4941   class class class wbr 5093   × cxp 5617  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  Oncon0 6311  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7352  ωcom 7802  1^st c1st 7925  2^nd c2nd 7926   ≈ cen 8872   ≼ cdom 8873  cardccrd 9835  ℝcr 11012  0cc0 11013  ℝ^*cxr 11152   ≤ cle 11154  ℕcn 12132  (,)cioo 13247  [,]cicc 13250  vol*covol 25391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-inf 9334  df-oi 9403  df-dju 9801  df-card 9839  df-acn 9842  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-q 12849  df-rp 12893  df-xadd 13014  df-ioo 13251  df-ico 13253  df-icc 13254  df-fz 13410  df-fzo 13557  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-sum 15596  df-xmet 21286  df-met 21287  df-ovol 25393

This theorem is referenced by:  uniioombllem3  25514  uniioombllem4  25515  uniioombllem5  25516  uniiccmbl  25519