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Theorem uniiccdif 25929

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 4179	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 25817	. . . . . . . 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 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
5		rexr 11382	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ∈ ℝ → (1^st ‘(𝐹‘𝑥)) ∈ ℝ^*)
6		rexr 11382	. . . . . . . 8 ⊢ ((2^nd ‘(𝐹‘𝑥)) ∈ ℝ → (2^nd ‘(𝐹‘𝑥)) ∈ ℝ^*)
7		id 23	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)) → (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)))
8		prunioo 13639	. . . . . . . 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 1178	. . . . . . 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 18	. . . . . 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 7008	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelcdmda 7108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 8064	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
16	14, 15	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
17	16	fveq2d 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7446	. . . . . . . . 9 ⊢ ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2816	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
21		df-pr 4591	. . . . . . . 8 ⊢ {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})
22		fvco3 7008	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
23	2, 22	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
24		fvco3 7008	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4702	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2812	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4171	. . . . . 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 7008	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6912	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7446	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2809	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 4979	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
37		iccf 13606	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
38		ffn 6731	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ^* × ℝ^*)
40		inss2 4137	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 11381	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
42	40, 41	sstri 3922	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
43		fss 6748	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
44	2, 42, 43	sylancl 598	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
45		fnfco 6769	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 599	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7274	. . . . 5 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
48	46, 47	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
49		iunun 5056	. . . . 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 13605	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
51		ffn 6731	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ^* × ℝ^*)
53		fnfco 6769	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 599	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7274	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
56	54, 55	syl 18	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
57		iunun 5056	. . . . . . 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 8045	. . . . . . . . . . . . . 14 ⊢ 1^st :V⟶ontoV
59		fofn 6820	. . . . . . . . . . . . . 14 ⊢ (1^st :V⟶ontoV → 1^st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1^st Fn V
61		ssv 3937	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6748	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6769	. . . . . . . . . . . . 13 ⊢ ((1^st Fn V ∧ 𝐹:ℕ⟶V) → (1^st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ∘ 𝐹) Fn ℕ)
66		fnfun 6661	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → Fun (1^st ∘ 𝐹))
67	65, 66	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1^st ∘ 𝐹))
68		fndm 6664	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → dom (1^st ∘ 𝐹) = ℕ)
69		eqimss2 3972	. . . . . . . . . . . 12 ⊢ (dom (1^st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1^st ∘ 𝐹))
70	65, 68, 69	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1^st ∘ 𝐹))
71		dfimafn2 6971	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)})
73		fnima 6691	. . . . . . . . . . 11 ⊢ ((1^st ∘ 𝐹) Fn ℕ → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
74	65, 73	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
75	72, 74	eqtr3d 2800	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = ran (1^st ∘ 𝐹))
76		rnco2 6268	. . . . . . . . 9 ⊢ ran (1^st ∘ 𝐹) = (1^st “ ran 𝐹)
77	75, 76	eqtrdi 2814	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = (1^st “ ran 𝐹))
78		fo2nd 8046	. . . . . . . . . . . . . 14 ⊢ 2^nd :V⟶ontoV
79		fofn 6820	. . . . . . . . . . . . . 14 ⊢ (2^nd :V⟶ontoV → 2^nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2^nd Fn V
81		fnfco 6769	. . . . . . . . . . . . 13 ⊢ ((2^nd Fn V ∧ 𝐹:ℕ⟶V) → (2^nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ∘ 𝐹) Fn ℕ)
83		fnfun 6661	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → Fun (2^nd ∘ 𝐹))
84	82, 83	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2^nd ∘ 𝐹))
85		fndm 6664	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → dom (2^nd ∘ 𝐹) = ℕ)
86		eqimss2 3972	. . . . . . . . . . . 12 ⊢ (dom (2^nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2^nd ∘ 𝐹))
87	82, 85, 86	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2^nd ∘ 𝐹))
88		dfimafn2 6971	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
90		fnima 6691	. . . . . . . . . . 11 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
91	82, 90	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
92	89, 91	eqtr3d 2800	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = ran (2^nd ∘ 𝐹))
93		rnco2 6268	. . . . . . . . 9 ⊢ ran (2^nd ∘ 𝐹) = (2^nd “ ran 𝐹)
94	92, 93	eqtrdi 2814	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = (2^nd “ ran 𝐹))
95	77, 94	uneq12d 4171	. . . . . . 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 2810	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
97	56, 96	uneq12d 4171	. . . . 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 2810	. . . 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 2806	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
100	1, 99	sseqtrrid 3956	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 25820	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 18	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4086	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9674	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 14150	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 9059	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 10054	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 705	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6819	. . . . . . . . . . . . 13 ⊢ (1^st :V⟶ontoV → Fun 1^st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1^st
111		ssv 3937	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6818	. . . . . . . . . . . . . . 15 ⊢ (1^st :V⟶ontoV → 1^st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1^st :V⟶V
114	113	fdmi 6743	. . . . . . . . . . . . 13 ⊢ dom 1^st = V
115	111, 114	sseqtrri 3962	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1^st
116		fores 6828	. . . . . . . . . . . 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 705	. . . . . . . . . . 11 ⊢ (1^st ↾ ran 𝐹):ran 𝐹⟶onto(1^st “ ran 𝐹)
118	2	ffnd 6732	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6824	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ⟶ontoran 𝐹)
120	118, 119	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ontoran 𝐹)
121		foco 6832	. . . . . . . . . . 11 ⊢ (((1^st ↾ ran 𝐹):ran 𝐹⟶onto(1^st “ ran 𝐹) ∧ 𝐹:ℕ⟶ontoran 𝐹) → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ⟶onto(1^st “ ran 𝐹))
122	117, 120, 121	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ⟶onto(1^st “ ran 𝐹))
123		fodomnum 10163	. . . . . . . . . 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 69	. . . . . . . . 9 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ℕ)
125		domentr 9068	. . . . . . . . 9 ⊢ (((1^st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1^st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ω)
127		fofun 6819	. . . . . . . . . . . . 13 ⊢ (2^nd :V⟶ontoV → Fun 2^nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2^nd
129		fof 6818	. . . . . . . . . . . . . . 15 ⊢ (2^nd :V⟶ontoV → 2^nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2^nd :V⟶V
131	130	fdmi 6743	. . . . . . . . . . . . 13 ⊢ dom 2^nd = V
132	111, 131	sseqtrri 3962	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2^nd
133		fores 6828	. . . . . . . . . . . 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 705	. . . . . . . . . . 11 ⊢ (2^nd ↾ ran 𝐹):ran 𝐹⟶onto(2^nd “ ran 𝐹)
135		foco 6832	. . . . . . . . . . 11 ⊢ (((2^nd ↾ ran 𝐹):ran 𝐹⟶onto(2^nd “ ran 𝐹) ∧ 𝐹:ℕ⟶ontoran 𝐹) → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ⟶onto(2^nd “ ran 𝐹))
136	134, 120, 135	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ⟶onto(2^nd “ ran 𝐹))
137		fodomnum 10163	. . . . . . . . . 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 69	. . . . . . . . 9 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ℕ)
139		domentr 9068	. . . . . . . . 9 ⊢ (((2^nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2^nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ω)
141		unctb 10309	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω)
143		ctex 9018	. . . . . . 7 ⊢ (((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
144	142, 143	syl 18	. . . . . 6 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
145		ssid 3935	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 3955	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
147		ssundif 4446	. . . . . . 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 221	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
149		ssdomg 9055	. . . . . 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 66	. . . . 5 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
151		domtr 9062	. . . . 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 596	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 9068	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 598	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 25843	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 596	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 521	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 𝒫 cpw 4560 {csn 4588 {cpr 4590 ⟨cop 4838 ∪ cuni 4870 ∪ ciun 4954 class class class wbr 5106 × cxp 5653 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 ∘ ccom 5659 Oncon0 6380 Fun wfun 6552 Fn wfn 6553 ⟶wf 6554 ⟶ontowfo 6556 ‘cfv 6558 (class class class)co 7443 ωcom 7901 1^st c1st 8023 2^nd c2nd 8024 ≈ cen 8998 ≼ cdom 8999 cardccrd 10043 ℝcr 11226 0cc0 11227 ℝ^*cxr 11369 ≤ cle 11371 ℕcn 12362 (,)cioo 13503 [,]cicc 13506 vol*covol 25813

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-sup 9461 df-inf 9462 df-oi 9531 df-dju 10009 df-card 10047 df-acn 10050 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-q 13103 df-rp 13148 df-xadd 13269 df-ioo 13507 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-sum 15883 df-xmet 21701 df-met 21702 df-ovol 25815

This theorem is used by: uniioombllem3 25936 uniioombllem4 25937 uniioombllem5 25938 uniiccmbl 25941

W3C validator