Theorem i1fd 25580

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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3		ffun 6655	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹)
4		funcnvcnv 6549	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
5		imadif 6566	. . . . . . . 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 13350	. . . . . . . . . . . . 13 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8		frn 6659	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran (,) ⊆ 𝒫 ℝ
10	9	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
11	10	elpwid 4560	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
12	11	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13		dfss4 4220	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
14	12, 13	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
15	14	imaeq2d 6011	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
16	6, 15	eqtr3d 2766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
17		fimacnv 6674	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶ℝ → (◡𝐹 “ ℝ) = ℝ)
18	2, 17	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) = ℝ)
19		rembl 25439	. . . . . . . 8 ⊢ ℝ ∈ dom vol
20	18, 19	eqeltrdi 2836	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) ∈ dom vol)
21	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22		inpreima 6998	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)))
23		iunid 5009	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
24	23	imaeq2i 6009	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (◡𝐹 “ (𝑦 ∩ ran 𝐹))
25		imaiun 7181	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
26	24, 25	eqtr3i 2754	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
27		cnvimass 6033	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
28		cnvimarndm 6034	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
29	27, 28	sseqtrri 3985	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹)
30		dfss2 3921	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦))
31	29, 30	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦)
32	22, 26, 31	3eqtr3g 2787	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36		inss2 4189	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37		ssfi 9087	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
38	35, 36, 37	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40		elinel1 4152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
41	40	con3i 154	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
42	41	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43		disjsn 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
44	42, 43	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45		reldisj 4404	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
48	47	sselda 3935	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (◡𝐹 “ {𝑥}) ∈ dom vol)
51	50	ralrimiva 3121	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
52		finiunmbl 25443	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
54	33, 53	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ∈ dom vol)
55	54	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
56	55	alrimiv 1927	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
57	56	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
58		elndif 4084	. . . . . . . . 9 ⊢ (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
59	58	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60		reex 11100	. . . . . . . . . 10 ⊢ ℝ ∈ V
61	60	difexi 5269	. . . . . . . . 9 ⊢ (ℝ ∖ 𝑥) ∈ V
62		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
63	62	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64		imaeq2 6007	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ 𝑥)))
65	64	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
66	63, 65	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
67	61, 66	spcv 3560	. . . . . . . 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 25442	. . . . . . 7 ⊢ (((◡𝐹 “ ℝ) ∈ dom vol ∧ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
70	20, 68, 69	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
71	16, 70	eqeltrrd 2829	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
72		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
73	72	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74		imaeq2 6007	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ 𝑦) = (◡𝐹 “ 𝑥))
75	74	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ 𝑥) ∈ dom vol))
76	73, 75	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol)))
77	76	spvv 1988	. . . . . . . 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 406	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
80	79	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
81	71, 80	pm2.61dan 812	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol)
82	81	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)
83		ismbf 25527	. . . 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 257	. 2 ⊢ (𝜑 → 𝐹 ∈ MblFn)
86		mblvol 25429	. . . . . . . 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 25430	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑦) ∈ dom vol → (◡𝐹 “ 𝑦) ⊆ ℝ)
89	54, 88	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ⊆ ℝ)
90		mblvol 25429	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)
94	91, 93	eqeltrrd 2829	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
95	38, 94	fsumrecl 15641	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
96	33	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ 𝑦)))
97		mblss 25430	. . . . . . . . . . . . 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 511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ))
100	99	ralrimiva 3121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ))
101		ovolfiniun 25400	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
103	96, 102	eqbrtrrd 5116	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
104		ovollecl 25382	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ∈ ℝ)
106	87, 105	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ)
107	106	ex 412	. . . . 5 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
108	107	alrimiv 1927	. . . 4 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
109		neldifsn 4743	. . . 4 ⊢ ¬ 0 ∈ (ℝ ∖ {0})
110	60	difexi 5269	. . . . 5 ⊢ (ℝ ∖ {0}) ∈ V
111		eleq2 2817	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112	111	notbid 318	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113		imaeq2 6007	. . . . . . . 8 ⊢ (𝑦 = (ℝ ∖ {0}) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ {0})))
114	113	fveq2d 6826	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (vol‘(◡𝐹 “ 𝑦)) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
115	114	eleq1d 2813	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → ((vol‘(◡𝐹 “ 𝑦)) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
116	112, 115	imbi12d 344	. . . . 5 ⊢ (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
117	110, 116	spcv 3560	. . . 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 1128	. 2 ⊢ (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
120		isi1f 25573	. 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
121	85, 119, 120	sylanbrc 583	1 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ ciun 4941   class class class wbr 5092   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  Fun wfun 6476  ⟶wf 6478  ‘cfv 6482  Fincfn 8872  ℝcr 11008  0cc0 11009  ℝ^*cxr 11148   ≤ cle 11150  (,)cioo 13248  Σcsu 15593  vol*covol 25361  volcvol 25362  MblFncmbf 25513  ∫₁citg1 25514

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xadd 13015  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-xmet 21254  df-met 21255  df-ovol 25363  df-vol 25364  df-mbf 25518  df-itg1 25519

This theorem is referenced by:  i1f0  25586  i1f1  25589  i1fadd  25594  i1fmul  25595  i1fmulc  25602  i1fres  25604  mbfi1fseqlem4  25617  itg2addnclem2  37652  ftc1anclem3  37675