Theorem i1fd 25740

Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fd.1	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
i1fd.2	⊢ (𝜑 → ran 𝐹 ∈ Fin)
i1fd.3	⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol)
i1fd.4	⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)

Assertion

Ref	Expression
i1fd	⊢ (𝜑 → 𝐹 ∈ dom ∫1)

Distinct variable groups:   𝑥,𝐹   𝜑,𝑥

Proof of Theorem i1fd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2	1	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3		ffun 6694	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹)
4		funcnvcnv 6588	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
5		imadif 6605	. . . . . . . 8 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))))
6	2, 3, 4, 5	4syl 19	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))))
7		ioof 13451	. . . . . . . . . . . . 13 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8		frn 6699	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran (,) ⊆ 𝒫 ℝ
10	9	sseli 3932	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
11	10	elpwid 4564	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
12	11	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13		dfss4 4221	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
14	12, 13	sylib 220	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
15	14	imaeq2d 6049	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
16	6, 15	eqtr3d 2799	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
17		fimacnv 6714	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶ℝ → (◡𝐹 “ ℝ) = ℝ)
18	2, 17	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) = ℝ)
19		rembl 25599	. . . . . . . 8 ⊢ ℝ ∈ dom vol
20	18, 19	eqeltrdi 2870	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) ∈ dom vol)
21	1	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22		inpreima 7045	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)))
23		iunid 5018	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
24	23	imaeq2i 6047	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (◡𝐹 “ (𝑦 ∩ ran 𝐹))
25		imaiun 7229	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
26	24, 25	eqtr3i 2787	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
27		cnvimass 6071	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
28		cnvimarndm 6072	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
29	27, 28	sseqtrri 3985	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹)
30		dfss2 3922	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦))
31	29, 30	mpbi 232	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦)
32	22, 26, 31	3eqtr3g 2820	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) = (◡𝐹 “ 𝑦))
33	21, 3, 32	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) = (◡𝐹 “ 𝑦))
34		i1fd.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
35	34	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36		inss2 4189	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37		ssfi 9141	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
38	35, 36, 37	sylancl 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39		simpll 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40		elinel1 4153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
41	40	con3i 154	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
42	41	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43		disjsn 4670	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
44	42, 43	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45		reldisj 4407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ ran 𝐹) ⊆ ran 𝐹 → (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0})))
46	36, 45	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
47	44, 46	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
48	47	sselda 3936	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝑥 ∈ (ran 𝐹 ∖ {0}))
49		i1fd.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol)
50	39, 48, 49	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (◡𝐹 “ {𝑥}) ∈ dom vol)
51	50	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
52		finiunmbl 25603	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
53	38, 51, 52	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
54	33, 53	eqeltrrd 2863	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ∈ dom vol)
55	54	ex 416	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
56	55	alrimiv 1947	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
57	56	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
58		elndif 4086	. . . . . . . . 9 ⊢ (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
59	58	adantl 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60		reex 11164	. . . . . . . . . 10 ⊢ ℝ ∈ V
61	60	difexi 5286	. . . . . . . . 9 ⊢ (ℝ ∖ 𝑥) ∈ V
62		eleq2 2851	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
63	62	notbid 320	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64		imaeq2 6045	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ 𝑥)))
65	64	eleq1d 2847	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
66	63, 65	imbi12d 346	. . . . . . . . 9 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
67	61, 66	spcv 3564	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
68	57, 59, 67	sylc 65	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
69		difmbl 25602	. . . . . . 7 ⊢ (((◡𝐹 “ ℝ) ∈ dom vol ∧ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
70	20, 68, 69	syl2anc 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
71	16, 70	eqeltrrd 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
72		eleq2 2851	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
73	72	notbid 320	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74		imaeq2 6045	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ 𝑦) = (◡𝐹 “ 𝑥))
75	74	eleq1d 2847	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ 𝑥) ∈ dom vol))
76	73, 75	imbi12d 346	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol)))
77	76	spvv 2008	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol))
78	56, 77	syl 17	. . . . . . 7 ⊢ (𝜑 → (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol))
79	78	imp 410	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
80	79	adantlr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
81	71, 80	pm2.61dan 822	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol)
82	81	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)
83		ismbf 25687	. . . 4 ⊢ (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
84	1, 83	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
85	82, 84	mpbird 259	. 2 ⊢ (𝜑 → 𝐹 ∈ MblFn)
86		mblvol 25589	. . . . . . . 8 ⊢ ((◡𝐹 “ 𝑦) ∈ dom vol → (vol‘(◡𝐹 “ 𝑦)) = (vol*‘(◡𝐹 “ 𝑦)))
87	54, 86	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(◡𝐹 “ 𝑦)) = (vol*‘(◡𝐹 “ 𝑦)))
88		mblss 25590	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑦) ∈ dom vol → (◡𝐹 “ 𝑦) ⊆ ℝ)
89	54, 88	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ⊆ ℝ)
90		mblvol 25589	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ {𝑥})))
91	50, 90	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ {𝑥})))
92		i1fd.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)
93	39, 48, 92	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)
94	91, 93	eqeltrrd 2863	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
95	38, 94	fsumrecl 15761	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
96	33	fveq2d 6871	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ 𝑦)))
97		mblss 25590	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (◡𝐹 “ {𝑥}) ⊆ ℝ)
98	50, 97	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (◡𝐹 “ {𝑥}) ⊆ ℝ)
99	98, 94	jca 519	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ))
100	99	ralrimiva 3154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ))
101		ovolfiniun 25560	. . . . . . . . . 10 ⊢ (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
102	38, 100, 101	syl2anc 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
103	96, 102	eqbrtrrd 5124	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
104		ovollecl 25542	. . . . . . . 8 ⊢ (((◡𝐹 “ 𝑦) ⊆ ℝ ∧ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ ∧ (vol*‘(◡𝐹 “ 𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥}))) → (vol*‘(◡𝐹 “ 𝑦)) ∈ ℝ)
105	89, 95, 103, 104	syl3anc 1390	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ∈ ℝ)
106	87, 105	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ)
107	106	ex 416	. . . . 5 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
108	107	alrimiv 1947	. . . 4 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
109		neldifsn 4752	. . . 4 ⊢ ¬ 0 ∈ (ℝ ∖ {0})
110	60	difexi 5286	. . . . 5 ⊢ (ℝ ∖ {0}) ∈ V
111		eleq2 2851	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112	111	notbid 320	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113		imaeq2 6045	. . . . . . . 8 ⊢ (𝑦 = (ℝ ∖ {0}) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ {0})))
114	113	fveq2d 6871	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (vol‘(◡𝐹 “ 𝑦)) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
115	114	eleq1d 2847	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → ((vol‘(◡𝐹 “ 𝑦)) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
116	112, 115	imbi12d 346	. . . . 5 ⊢ (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
117	110, 116	spcv 3564	. . . 4 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
118	108, 109, 117	mpisyl 21	. . 3 ⊢ (𝜑 → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
119	1, 34, 118	3jca 1141	. 2 ⊢ (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
120		isi1f 25733	. 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
121	85, 119, 120	sylanbrc 592	1 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  {csn 4582  ∪ ciun 4949   class class class wbr 5100   × cxp 5645  ^◡ccnv 5646  dom cdm 5647  ran crn 5648   “ cima 5650  Fun wfun 6515  ⟶wf 6517  ‘cfv 6521  Fincfn 8927  ℝcr 11072  0cc0 11073  ℝ^*cxr 11215   ≤ cle 11217  (,)cioo 13349  Σcsu 15713  vol*covol 25521  volcvol 25522  MblFncmbf 25673  ∫₁citg1 25674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-inf 9389  df-oi 9458  df-dju 9859  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-q 12950  df-rp 12994  df-xadd 13115  df-ioo 13353  df-ico 13355  df-icc 13356  df-fz 13513  df-fzo 13660  df-fl 13802  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-sum 15714  df-xmet 21414  df-met 21415  df-ovol 25523  df-vol 25524  df-mbf 25678  df-itg1 25679

This theorem is referenced by:  i1f0  25746  i1f1  25749  i1fadd  25754  i1fmul  25755  i1fmulc  25762  i1fres  25764  mbfi1fseqlem4  25777  itg2addnclem2  38168  ftc1anclem3  38191