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Theorem uniiccdif 25095

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 4173	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 24983	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
4	2, 3	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
5		rexr 11260	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ∈ ℝ → (1^st ‘(𝐹‘𝑥)) ∈ ℝ^*)
6		rexr 11260	. . . . . . . 8 ⊢ ((2^nd ‘(𝐹‘𝑥)) ∈ ℝ → (2^nd ‘(𝐹‘𝑥)) ∈ ℝ^*)
7		id 22	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)) → (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)))
8		prunioo 13458	. . . . . . . 8 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ^* ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
9	5, 6, 7, 8	syl3an 1161	. . . . . . 7 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
10	4, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
11		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelcdmda 7087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 8014	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
17	16	fveq2d 6896	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7412	. . . . . . . . 9 ⊢ ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2791	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
21		df-pr 4632	. . . . . . . 8 ⊢ {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})
22		fvco3 6991	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
23	2, 22	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
24		fvco3 6991	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4165	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}))
29		fvco3 6991	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7412	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2773	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 5022	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
37		iccf 13425	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
38		ffn 6718	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ^* × ℝ^*)
40		inss2 4230	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 11259	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
42	40, 41	sstri 3992	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
43		fss 6735	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
44	2, 42, 43	sylancl 587	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
45		fnfco 6757	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 588	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7246	. . . . 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 5097	. . . . 5 ⊢ ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}))
50		ioof 13424	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
51		ffn 6718	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ^* × ℝ^*)
53		fnfco 6757	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 588	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7246	. . . . . . 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 5097	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
58		fo1st 7995	. . . . . . . . . . . . . 14 ⊢ 1^st :V–onto→V
59		fofn 6808	. . . . . . . . . . . . . 14 ⊢ (1^st :V–onto→V → 1^st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1^st Fn V
61		ssv 4007	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6735	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6757	. . . . . . . . . . . . 13 ⊢ ((1^st Fn V ∧ 𝐹:ℕ⟶V) → (1^st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ∘ 𝐹) Fn ℕ)
66		fnfun 6650	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → Fun (1^st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1^st ∘ 𝐹))
68		fndm 6653	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → dom (1^st ∘ 𝐹) = ℕ)
69		eqimss2 4042	. . . . . . . . . . . 12 ⊢ (dom (1^st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1^st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1^st ∘ 𝐹))
71		dfimafn2 6956	. . . . . . . . . . 11 ⊢ ((Fun (1^st ∘ 𝐹) ∧ ℕ ⊆ dom (1^st ∘ 𝐹)) → ((1^st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)})
72	67, 70, 71	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)})
73		fnima 6681	. . . . . . . . . . 11 ⊢ ((1^st ∘ 𝐹) Fn ℕ → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
74	65, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
75	72, 74	eqtr3d 2775	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = ran (1^st ∘ 𝐹))
76		rnco2 6253	. . . . . . . . 9 ⊢ ran (1^st ∘ 𝐹) = (1^st “ ran 𝐹)
77	75, 76	eqtrdi 2789	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = (1^st “ ran 𝐹))
78		fo2nd 7996	. . . . . . . . . . . . . 14 ⊢ 2^nd :V–onto→V
79		fofn 6808	. . . . . . . . . . . . . 14 ⊢ (2^nd :V–onto→V → 2^nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2^nd Fn V
81		fnfco 6757	. . . . . . . . . . . . 13 ⊢ ((2^nd Fn V ∧ 𝐹:ℕ⟶V) → (2^nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ∘ 𝐹) Fn ℕ)
83		fnfun 6650	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → Fun (2^nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2^nd ∘ 𝐹))
85		fndm 6653	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → dom (2^nd ∘ 𝐹) = ℕ)
86		eqimss2 4042	. . . . . . . . . . . 12 ⊢ (dom (2^nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2^nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2^nd ∘ 𝐹))
88		dfimafn2 6956	. . . . . . . . . . 11 ⊢ ((Fun (2^nd ∘ 𝐹) ∧ ℕ ⊆ dom (2^nd ∘ 𝐹)) → ((2^nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
89	84, 87, 88	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
90		fnima 6681	. . . . . . . . . . 11 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
91	82, 90	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
92	89, 91	eqtr3d 2775	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = ran (2^nd ∘ 𝐹))
93		rnco2 6253	. . . . . . . . 9 ⊢ ran (2^nd ∘ 𝐹) = (2^nd “ ran 𝐹)
94	92, 93	eqtrdi 2789	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = (2^nd “ ran 𝐹))
95	77, 94	uneq12d 4165	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)}) = ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
96	57, 95	eqtrid 2785	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
97	56, 96	uneq12d 4165	. . . . 5 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
98	49, 97	eqtrid 2785	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
99	36, 48, 98	3eqtr3d 2781	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
100	1, 99	sseqtrrid 4036	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 24986	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4143	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9641	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13945	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 9000	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9941	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 691	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6807	. . . . . . . . . . . . 13 ⊢ (1^st :V–onto→V → Fun 1^st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1^st
111		ssv 4007	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6806	. . . . . . . . . . . . . . 15 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1^st :V⟶V
114	113	fdmi 6730	. . . . . . . . . . . . 13 ⊢ dom 1^st = V
115	111, 114	sseqtrri 4020	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1^st
116		fores 6816	. . . . . . . . . . . 12 ⊢ ((Fun 1^st ∧ ran 𝐹 ⊆ dom 1^st ) → (1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹))
117	110, 115, 116	mp2an 691	. . . . . . . . . . 11 ⊢ (1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹)
118	2	ffnd 6719	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6812	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6820	. . . . . . . . . . 11 ⊢ (((1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹))
122	117, 120, 121	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹))
123		fodomnum 10052	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹) → (1^st “ ran 𝐹) ≼ ℕ))
124	108, 122, 123	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ℕ)
125		domentr 9009	. . . . . . . . 9 ⊢ (((1^st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1^st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ω)
127		fofun 6807	. . . . . . . . . . . . 13 ⊢ (2^nd :V–onto→V → Fun 2^nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2^nd
129		fof 6806	. . . . . . . . . . . . . . 15 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2^nd :V⟶V
131	130	fdmi 6730	. . . . . . . . . . . . 13 ⊢ dom 2^nd = V
132	111, 131	sseqtrri 4020	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2^nd
133		fores 6816	. . . . . . . . . . . 12 ⊢ ((Fun 2^nd ∧ ran 𝐹 ⊆ dom 2^nd ) → (2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹))
134	128, 132, 133	mp2an 691	. . . . . . . . . . 11 ⊢ (2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹)
135		foco 6820	. . . . . . . . . . 11 ⊢ (((2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹))
136	134, 120, 135	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹))
137		fodomnum 10052	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹) → (2^nd “ ran 𝐹) ≼ ℕ))
138	108, 136, 137	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ℕ)
139		domentr 9009	. . . . . . . . 9 ⊢ (((2^nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2^nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ω)
141		unctb 10200	. . . . . . . 8 ⊢ (((1^st “ ran 𝐹) ≼ ω ∧ (2^nd “ ran 𝐹) ≼ ω) → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω)
142	126, 140, 141	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω)
143		ctex 8959	. . . . . . 7 ⊢ (((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
144	142, 143	syl 17	. . . . . 6 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
145		ssid 4005	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 4035	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
147		ssundif 4488	. . . . . . 7 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))) ↔ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
148	146, 147	sylib 217	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
149		ssdomg 8996	. . . . . 6 ⊢ (((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V → ((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
150	144, 148, 149	sylc 65	. . . . 5 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
151		domtr 9003	. . . . 5 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∧ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
152	150, 142, 151	syl2anc 585	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 9009	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 587	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 25009	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 585	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 513	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 {csn 4629 {cpr 4631 ⟨cop 4635 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 × cxp 5675 dom cdm 5677 ran crn 5678 ↾ cres 5679 “ cima 5680 ∘ ccom 5681 Oncon0 6365 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 ωcom 7855 1^st c1st 7973 2^nd c2nd 7974 ≈ cen 8936 ≼ cdom 8937 cardccrd 9930 ℝcr 11109 0cc0 11110 ℝ^*cxr 11247 ≤ cle 11249 ℕcn 12212 (,)cioo 13324 [,]cicc 13327 vol*covol 24979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-xmet 20937 df-met 20938 df-ovol 24981

This theorem is referenced by: uniioombllem3 25102 uniioombllem4 25103 uniioombllem5 25104 uniiccmbl 25107

W3C validator