Theorem i1fd 25909

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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3		ffun 6705	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹)
4		funcnvcnv 6600	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
5		imadif 6617	. . . . . . . 8 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))))
6	2, 3, 4, 5	4syl 20	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))))
7		ioof 13500	. . . . . . . . . . . . 13 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8		frn 6710	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran (,) ⊆ 𝒫 ℝ
10	9	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
11	10	elpwid 4566	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
12	11	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13		dfss4 4215	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
14	12, 13	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
15	14	imaeq2d 6056	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
16	6, 15	eqtr3d 2797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) = (◡𝐹 “ 𝑥))
17		fimacnv 6725	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶ℝ → (◡𝐹 “ ℝ) = ℝ)
18	2, 17	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) = ℝ)
19		rembl 25768	. . . . . . . 8 ⊢ ℝ ∈ dom vol
20	18, 19	eqeltrdi 2868	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ ℝ) ∈ dom vol)
21	1	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22		inpreima 7056	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)))
23		iunid 5019	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
24	23	imaeq2i 6054	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (◡𝐹 “ (𝑦 ∩ ran 𝐹))
25		imaiun 7242	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
26	24, 25	eqtr3i 2785	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝑦 ∩ ran 𝐹)) = ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})
27		cnvimass 6078	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
28		cnvimarndm 6079	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
29	27, 28	sseqtrri 3980	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹)
30		dfss2 3917	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐹 “ 𝑦) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦))
31	29, 30	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝑦)
32	22, 26, 31	3eqtr3g 2818	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) = (◡𝐹 “ 𝑦))
33	21, 3, 32	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) = (◡𝐹 “ 𝑦))
34		i1fd.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
35	34	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36		inss2 4183	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37		ssfi 9167	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
38	35, 36, 37	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40		elinel1 4147	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
41	40	con3i 155	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
42	41	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43		disjsn 4672	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
44	42, 43	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45		reldisj 4406	. . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
48	47	sselda 3931	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 25772	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥}) ∈ dom vol)
54	33, 53	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ∈ dom vol)
55	54	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
56	55	alrimiv 1960	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
57	56	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol))
58		elndif 4080	. . . . . . . . 9 ⊢ (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
59	58	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60		reex 11215	. . . . . . . . . 10 ⊢ ℝ ∈ V
61	60	difexi 5295	. . . . . . . . 9 ⊢ (ℝ ∖ 𝑥) ∈ V
62		eleq2 2849	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
63	62	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64		imaeq2 6052	. . . . . . . . . . 11 ⊢ (𝑦 = (ℝ ∖ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ 𝑥)))
65	64	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
66	63, 65	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
67	61, 66	spcv 3559	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
68	57, 59, 67	sylc 66	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
69		difmbl 25771	. . . . . . 7 ⊢ (((◡𝐹 “ ℝ) ∈ dom vol ∧ (◡𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
70	20, 68, 69	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((◡𝐹 “ ℝ) ∖ (◡𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
71	16, 70	eqeltrrd 2861	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
72		eleq2 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
73	72	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74		imaeq2 6052	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ 𝑦) = (◡𝐹 “ 𝑥))
75	74	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((◡𝐹 “ 𝑦) ∈ dom vol ↔ (◡𝐹 “ 𝑥) ∈ dom vol))
76	73, 75	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol)))
77	76	spvv 2021	. . . . . . . 8 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (◡𝐹 “ 𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol))
78	56, 77	syl 18	. . . . . . 7 ⊢ (𝜑 → (¬ 0 ∈ 𝑥 → (◡𝐹 “ 𝑥) ∈ dom vol))
79	78	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
80	79	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (◡𝐹 “ 𝑥) ∈ dom vol)
81	71, 80	pm2.61dan 825	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol)
82	81	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)
83		ismbf 25856	. . . 4 ⊢ (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
84	1, 83	syl 18	. . 3 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol))
85	82, 84	mpbird 260	. 2 ⊢ (𝜑 → 𝐹 ∈ MblFn)
86		mblvol 25758	. . . . . . . 8 ⊢ ((◡𝐹 “ 𝑦) ∈ dom vol → (vol‘(◡𝐹 “ 𝑦)) = (vol*‘(◡𝐹 “ 𝑦)))
87	54, 86	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(◡𝐹 “ 𝑦)) = (vol*‘(◡𝐹 “ 𝑦)))
88		mblss 25759	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑦) ∈ dom vol → (◡𝐹 “ 𝑦) ⊆ ℝ)
89	54, 88	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (◡𝐹 “ 𝑦) ⊆ ℝ)
90		mblvol 25758	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ {𝑥})))
91	50, 90	syl 18	. . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)
94	91, 93	eqeltrrd 2861	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
95	38, 94	fsumrecl 15820	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})) ∈ ℝ)
96	33	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ 𝑦)))
97		mblss 25759	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (◡𝐹 “ {𝑥}) ⊆ ℝ)
98	50, 97	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (◡𝐹 “ {𝑥}) ⊆ ℝ)
99	98, 94	jca 521	. . . . . . . . . . 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 25729	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘∪ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(◡𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
103	96, 102	eqbrtrrd 5129	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(◡𝐹 “ {𝑥})))
104		ovollecl 25711	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(◡𝐹 “ 𝑦)) ∈ ℝ)
106	87, 105	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ)
107	106	ex 418	. . . . 5 ⊢ (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
108	107	alrimiv 1960	. . . 4 ⊢ (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ))
109		neldifsn 4755	. . . 4 ⊢ ¬ 0 ∈ (ℝ ∖ {0})
110	60	difexi 5295	. . . . 5 ⊢ (ℝ ∖ {0}) ∈ V
111		eleq2 2849	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112	111	notbid 321	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113		imaeq2 6052	. . . . . . . 8 ⊢ (𝑦 = (ℝ ∖ {0}) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (ℝ ∖ {0})))
114	113	fveq2d 6882	. . . . . . 7 ⊢ (𝑦 = (ℝ ∖ {0}) → (vol‘(◡𝐹 “ 𝑦)) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
115	114	eleq1d 2845	. . . . . 6 ⊢ (𝑦 = (ℝ ∖ {0}) → ((vol‘(◡𝐹 “ 𝑦)) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
116	112, 115	imbi12d 347	. . . . 5 ⊢ (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
117	110, 116	spcv 3559	. . . 4 ⊢ (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(◡𝐹 “ 𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
118	108, 109, 117	mpisyl 22	. . 3 ⊢ (𝜑 → (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
119	1, 34, 118	3jca 1146	. 2 ⊢ (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
120		isi1f 25902	. 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
121	85, 119, 120	sylanbrc 595	1 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ ciun 4951   class class class wbr 5103   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  Fun wfun 6527  ⟶wf 6529  ‘cfv 6533  Fincfn 8952  ℝcr 11123  0cc0 11124  ℝ^*cxr 11266   ≤ cle 11268  (,)cioo 13398  Σcsu 15773  vol*covol 25690  volcvol 25691  MblFncmbf 25842  ∫₁citg1 25843

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-q 12998  df-rp 13043  df-xadd 13164  df-ioo 13402  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-xmet 21578  df-met 21579  df-ovol 25692  df-vol 25693  df-mbf 25847  df-itg1 25848

This theorem is used by:  i1f0  25915  i1f1  25918  i1fadd  25923  i1fmul  25924  i1fmulc  25931  i1fres  25933  mbfi1fseqlem4  25946  itg2addnclem2  38421  ftc1anclem3  38444