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Theorem i1fd 23448

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 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3		ffun 6048	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹)
4		funcnvcnv 5956	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ^◡^◡𝐹)
5		imadif 5973	. . . . . . . 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 12271	. . . . . . . . . . . . 13 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
8		frn 6053	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran (,) ⊆ 𝒫 ℝ
10	9	sseli 3599	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
11	10	elpwid 4170	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
12	11	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13		dfss4 3858	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
14	12, 13	sylib 208	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
15	14	imaeq2d 5466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (^◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (^◡𝐹 “ 𝑥))
16	6, 15	eqtr3d 2658	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ (ℝ ∖ 𝑥))) = (^◡𝐹 “ 𝑥))
17		fimacnv 6347	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶ℝ → (^◡𝐹 “ ℝ) = ℝ)
18	2, 17	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (^◡𝐹 “ ℝ) = ℝ)
19		rembl 23308	. . . . . . . 8 ⊢ ℝ ∈ dom vol
20	18, 19	syl6eqel 2709	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (^◡𝐹 “ ℝ) ∈ dom vol)
21	1	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22		inpreima 6342	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ((^◡𝐹 “ 𝑦) ∩ (^◡𝐹 “ ran 𝐹)))
23		iunid 4575	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
24	23	imaeq2i 5464	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (^◡𝐹 “ (𝑦 ∩ ran 𝐹))
25		imaiun 6503	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥})
26	24, 25	eqtr3i 2646	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥})
27		cnvimass 5485	. . . . . . . . . . . . . . . 16 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
28		cnvimarndm 5486	. . . . . . . . . . . . . . . 16 ⊢ (^◡𝐹 “ ran 𝐹) = dom 𝐹
29	27, 28	sseqtr4i 3638	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ 𝑦) ⊆ (^◡𝐹 “ ran 𝐹)
30		df-ss 3588	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹 “ 𝑦) ⊆ (^◡𝐹 “ ran 𝐹) ↔ ((^◡𝐹 “ 𝑦) ∩ (^◡𝐹 “ ran 𝐹)) = (^◡𝐹 “ 𝑦))
31	29, 30	mpbi 220	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹 “ 𝑦) ∩ (^◡𝐹 “ ran 𝐹)) = (^◡𝐹 “ 𝑦)
32	22, 26, 31	3eqtr3g 2679	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36		inss2 3834	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37		ssfi 8180	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
38	35, 36, 37	sylancl 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39		simpll 790	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40		inss1 3833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ ran 𝐹) ⊆ 𝑦
41	40	sseli 3599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
42	41	con3i 150	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43	42	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
44		disjsn 4246	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
45	43, 44	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
46		reldisj 4020	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ ran 𝐹) ⊆ ran 𝐹 → (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0})))
47	36, 46	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
48	45, 47	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
49	48	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝑥 ∈ (ran 𝐹 ∖ {0}))
50		i1fd.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (^◡𝐹 “ {𝑥}) ∈ dom vol)
51	39, 49, 50	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (^◡𝐹 “ {𝑥}) ∈ dom vol)
52	51	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥}) ∈ dom vol)
53		finiunmbl 23312	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥}) ∈ dom vol) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥}) ∈ dom vol)
54	38, 52, 53	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥}) ∈ dom vol)
55	33, 54	eqeltrrd 2702	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ∈ dom vol)
56	55	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol))
57	56	alrimiv 1855	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol))
58	57	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol))
59		elndif 3734	. . . . . . . . 9 ⊢ (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
60	59	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
61		reex 10027	. . . . . . . . . 10 ⊢ ℝ ∈ V
62		difexg 4808	. . . . . . . . . 10 ⊢ (ℝ ∈ V → (ℝ ∖ 𝑥) ∈ V)
63	61, 62	ax-mp 5	. . . . . . . . 9 ⊢ (ℝ ∖ 𝑥) ∈ V
64		eleq2 2690	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
65	64	notbid 308	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
66		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ (ℝ ∖ 𝑥)))
67	66	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((^◡𝐹 “ 𝑦) ∈ dom vol ↔ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
68	65, 67	imbi12d 334	. . . . . . . . 9 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
69	63, 68	spcv 3299	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
70	58, 60, 69	sylc 65	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
71		difmbl 23311	. . . . . . 7 ⊢ (((^◡𝐹 “ ℝ) ∈ dom vol ∧ (^◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
72	20, 70, 71	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
73	16, 72	eqeltrrd 2702	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (^◡𝐹 “ 𝑥) ∈ dom vol)
74		eleq2 2690	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
75	74	notbid 308	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
76		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ 𝑥))
77	76	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((^◡𝐹 “ 𝑦) ∈ dom vol ↔ (^◡𝐹 “ 𝑥) ∈ dom vol))
78	75, 77	imbi12d 334	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (^◡𝐹 “ 𝑥) ∈ dom vol)))
79	78	spv 2260	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (^◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (^◡𝐹 “ 𝑥) ∈ dom vol))
80	57, 79	syl 17	. . . . . . 7 ⊢ (𝜑 → (¬ 0 ∈ 𝑥 → (^◡𝐹 “ 𝑥) ∈ dom vol))
81	80	imp 445	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (^◡𝐹 “ 𝑥) ∈ dom vol)
82	81	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (^◡𝐹 “ 𝑥) ∈ dom vol)
83	73, 82	pm2.61dan 832	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (^◡𝐹 “ 𝑥) ∈ dom vol)
84	83	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(^◡𝐹 “ 𝑥) ∈ dom vol)
85		ismbf 23397	. . . 4 ⊢ (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡𝐹 “ 𝑥) ∈ dom vol))
86	1, 85	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡𝐹 “ 𝑥) ∈ dom vol))
87	84, 86	mpbird 247	. 2 ⊢ (𝜑 → 𝐹 ∈ MblFn)
88		mblvol 23298	. . . . . . . 8 ⊢ ((^◡𝐹 “ 𝑦) ∈ dom vol → (vol‘(^◡𝐹 “ 𝑦)) = (vol*‘(^◡𝐹 “ 𝑦)))
89	55, 88	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(^◡𝐹 “ 𝑦)) = (vol*‘(^◡𝐹 “ 𝑦)))
90		mblss 23299	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑦) ∈ dom vol → (^◡𝐹 “ 𝑦) ⊆ ℝ)
91	55, 90	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (^◡𝐹 “ 𝑦) ⊆ ℝ)
92		mblvol 23298	. . . . . . . . . . 11 ⊢ ((^◡𝐹 “ {𝑥}) ∈ dom vol → (vol‘(^◡𝐹 “ {𝑥})) = (vol*‘(^◡𝐹 “ {𝑥})))
93	51, 92	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(^◡𝐹 “ {𝑥})) = (vol*‘(^◡𝐹 “ {𝑥})))
94		i1fd.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(^◡𝐹 “ {𝑥})) ∈ ℝ)
95	39, 49, 94	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(^◡𝐹 “ {𝑥})) ∈ ℝ)
96	93, 95	eqeltrrd 2702	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(^◡𝐹 “ {𝑥})) ∈ ℝ)
97	38, 96	fsumrecl 14465	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(^◡𝐹 “ {𝑥})) ∈ ℝ)
98	33	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥})) = (vol‘(^◡𝐹 “ 𝑦)))
99		mblss 23299	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 “ {𝑥}) ∈ dom vol → (^◡𝐹 “ {𝑥}) ⊆ ℝ)
100	51, 99	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (^◡𝐹 “ {𝑥}) ⊆ ℝ)
101	100, 96	jca 554	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((^◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(^◡𝐹 “ {𝑥})) ∈ ℝ))
102	101	ralrimiva 2966	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((^◡𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(^◡𝐹 “ {𝑥})) ∈ ℝ))
103		ovolfiniun 23269	. . . . . . . . . 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})))
104	38, 102, 103	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(^◡𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol‘(^◡𝐹 “ {𝑥})))
105	98, 104	eqbrtrrd 4677	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(^◡𝐹 “ 𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol‘(^◡𝐹 “ {𝑥})))
106		ovollecl 23251	. . . . . . . 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)
107	91, 97, 105, 106	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(^◡𝐹 “ 𝑦)) ∈ ℝ)
108	89, 107	eqeltrd 2701	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ)
109	108	ex 450	. . . . 5 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ))
110	109	alrimiv 1855	. . . 4 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ))
111		neldifsn 4321	. . . 4 ⊢ ¬ 0 ∈ (ℝ ∖ {0})
112		difexg 4808	. . . . . 6 ⊢ (ℝ ∈ V → (ℝ ∖ {0}) ∈ V)
113	61, 112	ax-mp 5	. . . . 5 ⊢ (ℝ ∖ {0}) ∈ V
114		eleq2 2690	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
115	114	notbid 308	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
116		imaeq2 5462	. . . . . . . 8 ⊢ (𝑦 = (ℝ ∖ {0}) → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ (ℝ ∖ {0})))
117	116	fveq2d 6195	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (vol‘(^◡𝐹 “ 𝑦)) = (vol‘(^◡𝐹 “ (ℝ ∖ {0}))))
118	117	eleq1d 2686	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → ((vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ ↔ (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
119	115, 118	imbi12d 334	. . . . 5 ⊢ (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
120	113, 119	spcv 3299	. . . 4 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(^◡𝐹 “ 𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
121	110, 111, 120	mpisyl 21	. . 3 ⊢ (𝜑 → (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
122	1, 34, 121	3jca 1242	. 2 ⊢ (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
123		isi1f 23441	. 2 ⊢ (𝐹 ∈ dom ∫₁ ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(^◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
124	87, 122, 123	sylanbrc 698	1 ⊢ (𝜑 → 𝐹 ∈ dom ∫₁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ ciun 4520 class class class wbr 4653 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 Fincfn 7955 ℝcr 9935 0cc0 9936 ℝ^*cxr 10073 ≤ cle 10075 (,)cioo 12175 Σcsu 14416 vol*covol 23231 volcvol 23232 MblFncmbf 23383 ∫₁citg1 23384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389

This theorem is referenced by: i1f0 23454 i1f1 23457 i1fadd 23462 i1fmul 23463 i1fmulc 23470 i1fres 23472 mbfi1fseqlem4 23485 itg2addnclem2 33462 ftc1anclem3 33487

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