Theorem uniiccdif 25486

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 4144	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 25374	. . . . . . . 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 11227	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥)) ∈ ℝ → (1st ‘(𝐹‘𝑥)) ∈ ℝ*)
6		rexr 11227	. . . . . . . 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 13449	. . . . . . . 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 6963	. . . . . . . . 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 7059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4171	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 8010	. . . . . . . . . . 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 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7393	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
21		df-pr 4595	. . . . . . . 8 ⊢ {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})
22		fvco3 6963	. . . . . . . . . 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 6963	. . . . . . . . . 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 4708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)} = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = {(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4135	. . . . . 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 6963	. . . . . . . 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 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7393	. . . . . . . 8 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 4983	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)})))
37		iccf 13416	. . . . . . 7 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
38		ffn 6691	. . . . . . 7 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ* × ℝ*)
40		inss2 4204	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 11226	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42	40, 41	sstri 3959	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
43		fss 6707	. . . . . . 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 6728	. . . . . 6 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 587	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7224	. . . . 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 5060	. . . . 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 13415	. . . . . . . . 9 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
51		ffn 6691	. . . . . . . . 9 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ* × ℝ*)
53		fnfco 6728	. . . . . . . 8 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 587	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7224	. . . . . . 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 5060	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)})
58		fo1st 7991	. . . . . . . . . . . . . 14 ⊢ 1st :V–onto→V
59		fofn 6777	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st Fn V
61		ssv 3974	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6707	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6728	. . . . . . . . . . . . 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 6621	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → Fun (1st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1st ∘ 𝐹))
68		fndm 6624	. . . . . . . . . . . 12 ⊢ ((1st ∘ 𝐹) Fn ℕ → dom (1st ∘ 𝐹) = ℕ)
69		eqimss2 4009	. . . . . . . . . . . 12 ⊢ (dom (1st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1st ∘ 𝐹))
71		dfimafn2 6927	. . . . . . . . . . 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 6651	. . . . . . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = ran (1st ∘ 𝐹))
76		rnco2 6229	. . . . . . . . 9 ⊢ ran (1st ∘ 𝐹) = (1st “ ran 𝐹)
77	75, 76	eqtrdi 2781	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} = (1st “ ran 𝐹))
78		fo2nd 7992	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
79		fofn 6777	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → 2nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2nd Fn V
81		fnfco 6728	. . . . . . . . . . . . 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 6621	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → Fun (2nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2nd ∘ 𝐹))
85		fndm 6624	. . . . . . . . . . . 12 ⊢ ((2nd ∘ 𝐹) Fn ℕ → dom (2nd ∘ 𝐹) = ℕ)
86		eqimss2 4009	. . . . . . . . . . . 12 ⊢ (dom (2nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2nd ∘ 𝐹))
88		dfimafn2 6927	. . . . . . . . . . 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 6651	. . . . . . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = ran (2nd ∘ 𝐹))
93		rnco2 6229	. . . . . . . . 9 ⊢ ran (2nd ∘ 𝐹) = (2nd “ ran 𝐹)
94	92, 93	eqtrdi 2781	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)} = (2nd “ ran 𝐹))
95	77, 94	uneq12d 4135	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
96	57, 95	eqtrid 2777	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1st ∘ 𝐹)‘𝑥)} ∪ {((2nd ∘ 𝐹)‘𝑥)}) = ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹)))
97	56, 96	uneq12d 4135	. . . . 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 2777	. . . 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 2773	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
100	1, 99	sseqtrrid 3993	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 25377	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4113	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9606	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13952	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 8978	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9906	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 692	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6776	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
111		ssv 3974	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6775	. . . . . . . . . . . . . . 15 ⊢ (1st :V–onto→V → 1st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1st :V⟶V
114	113	fdmi 6702	. . . . . . . . . . . . 13 ⊢ dom 1st = V
115	111, 114	sseqtrri 3999	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1st
116		fores 6785	. . . . . . . . . . . 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 6692	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6781	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6789	. . . . . . . . . . 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 10017	. . . . . . . . . 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 8987	. . . . . . . . 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 6776	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
129		fof 6775	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
131	130	fdmi 6702	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
132	111, 131	sseqtrri 3999	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2nd
133		fores 6785	. . . . . . . . . . . 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 6789	. . . . . . . . . . 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 10017	. . . . . . . . . 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 8987	. . . . . . . . 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 10164	. . . . . . . 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 8938	. . . . . . 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 3972	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 3992	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1st “ ran 𝐹) ∪ (2nd “ ran 𝐹))))
147		ssundif 4454	. . . . . . 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 8974	. . . . . 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 8981	. . . . 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 8987	. . . 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 25400	. . 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 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  {cpr 4594  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5110   × cxp 5639  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644   ∘ ccom 5645  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390  ωcom 7845  1^st c1st 7969  2^nd c2nd 7970   ≈ cen 8918   ≼ cdom 8919  cardccrd 9895  ℝcr 11074  0cc0 11075  ℝ^*cxr 11214   ≤ cle 11216  ℕcn 12193  (,)cioo 13313  [,]cicc 13316  vol*covol 25370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-acn 9902  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xadd 13080  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660  df-xmet 21264  df-met 21265  df-ovol 25372

This theorem is referenced by:  uniioombllem3  25493  uniioombllem4  25494  uniioombllem5  25495  uniiccmbl  25498