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Theorem mbfposr 25551

Description: Converse to mbfpos 25550. (Contributed by Mario Carneiro, 11-Aug-2014.)

**Hypotheses**
Ref	Expression
mbfpos.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
mbfposr.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
mbfposr.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)

**Assertion**
Ref	Expression
mbfposr	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem mbfposr Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfpos.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
2	1	fmpttd 7049	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		mbfposr.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
4		0re 11117	. . . 4 ⊢ 0 ∈ ℝ
5		ifcl 4522	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
6	1, 4, 5	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
7	3, 6	mbfdm2 25536	. 2 ⊢ (𝜑 → 𝐴 ∈ dom vol)
8		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 < 0)
9		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
10	9	lt0neg1d 11689	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 0 ↔ 0 < -𝑦))
11	8, 10	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 0 < -𝑦)
12	11	biantrurd 532	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (-𝐵 < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
13	1	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	9, 13	ltnegd 11698	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ -𝐵 < -𝑦))
15		0red 11118	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
16	13	renegcld 11547	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
17	9	renegcld 11547	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
18		maxlt 13095	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ ∧ -𝑦 ∈ ℝ) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
19	15, 16, 17, 18	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
20	12, 14, 19	3bitr4rd 312	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ 𝑦 < 𝐵))
21	1	renegcld 11547	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
22		ifcl 4522	. . . . . . . . . . . . . 14 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
23	21, 4, 22	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
24	23	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
25	24	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) < -𝑦)))
26	13	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
27	20, 25, 26	3bitr3d 309	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) < -𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
28	17	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
29		elioomnf 13347	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ ℝ^* → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦) ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) < -𝑦)))
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦) ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) < -𝑦)))
31	9	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
32		elioopnf 13346	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
34	27, 30, 33	3bitr4d 311	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦) ↔ 𝐵 ∈ (𝑦(,)+∞)))
35		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
36		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
37	36	fvmpt2 6941	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) = if(0 ≤ -𝐵, -𝐵, 0))
38	35, 23, 37	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) = if(0 ≤ -𝐵, -𝐵, 0))
39	38	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦)))
40	39	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦)))
41		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
42	41	fvmpt2 6941	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
43	35, 1, 42	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
44	43	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
45	44	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
46	34, 40, 45	3bitr4d 311	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞)))
47	46	pm5.32da 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → ((𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
48	23	fmpttd 7049	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ)
49		ffn 6652	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) Fn 𝐴)
50		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦))))
51	48, 49, 50	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦))))
52	51	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦))))
53		ffn 6652	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
54		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
55	2, 53, 54	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
56	55	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
57	47, 52, 56	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
58	57	alrimiv 1927	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
59		nfmpt1 5191	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
60	59	nfcnv 5821	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
61		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(-∞(,)-𝑦)
62	60, 61	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦))
63		nfmpt1 5191	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
64	63	nfcnv 5821	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ 𝐵)
65		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦(,)+∞)
66	64, 65	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))
67	62, 66	cleqf 2920	. . . . 5 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
68	58, 67	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)))
69		mbfposr.3	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
70		mbfima 25529	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ∈ dom vol)
71	69, 48, 70	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ∈ dom vol)
72	71	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ∈ dom vol)
73	68, 72	eqeltrrd 2829	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
74		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝑦)
75		0red 11118	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
76	1	ad4ant14 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
77		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
78		maxle 13093	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ (0 ≤ 𝑦 ∧ 𝐵 ≤ 𝑦)))
79	75, 76, 77, 78	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ (0 ≤ 𝑦 ∧ 𝐵 ≤ 𝑦)))
80	74, 79	mpbirand 707	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
81	80	notbid 318	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (¬ if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ ¬ 𝐵 ≤ 𝑦))
82	76, 4, 5	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
83	77, 82	ltnled 11263	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < if(0 ≤ 𝐵, 𝐵, 0) ↔ ¬ if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦))
84	77, 76	ltnled 11263	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ ¬ 𝐵 ≤ 𝑦))
85	81, 83, 84	3bitr4d 311	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑦 < 𝐵))
86	82	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < if(0 ≤ 𝐵, 𝐵, 0) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 𝑦 < if(0 ≤ 𝐵, 𝐵, 0))))
87	76	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
88	85, 86, 87	3bitr3d 309	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 𝑦 < if(0 ≤ 𝐵, 𝐵, 0)) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
89	77	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
90		elioopnf 13346	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 𝑦 < if(0 ≤ 𝐵, 𝐵, 0))))
91	89, 90	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 𝑦 < if(0 ≤ 𝐵, 𝐵, 0))))
92	89, 32	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
93	88, 91, 92	3bitr4d 311	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
94		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
95	94	fvmpt2 6941	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) = if(0 ≤ 𝐵, 𝐵, 0))
96	35, 6, 95	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) = if(0 ≤ 𝐵, 𝐵, 0))
97	96	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞)))
98	97	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞)))
99	44	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
100	93, 98, 99	3bitr4d 311	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞)))
101	100	pm5.32da 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → ((𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
102	6	fmpttd 7049	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ)
103		ffn 6652	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) Fn 𝐴)
104		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞))))
105	102, 103, 104	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞))))
106	105	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞))))
107	55	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
108	101, 106, 107	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
109	108	alrimiv 1927	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
110		nfmpt1 5191	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
111	110	nfcnv 5821	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
112	111, 65	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞))
113	112, 66	cleqf 2920	. . . . 5 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
114	109, 113	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)))
115		mbfima 25529	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ∈ dom vol)
116	3, 102, 115	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ∈ dom vol)
117	116	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ∈ dom vol)
118	114, 117	eqeltrrd 2829	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
119		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
120		0red 11118	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℝ)
121	73, 118, 119, 120	ltlecasei 11224	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
122		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 < 𝑦)
123		0red 11118	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
124	1	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
125		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
126		maxlt 13095	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ (0 < 𝑦 ∧ 𝐵 < 𝑦)))
127	123, 124, 125, 126	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ (0 < 𝑦 ∧ 𝐵 < 𝑦)))
128	122, 127	mpbirand 707	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ 𝐵 < 𝑦))
129	6	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
130	129	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ if(0 ≤ 𝐵, 𝐵, 0) < 𝑦)))
131	124	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
132	128, 130, 131	3bitr3d 309	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ if(0 ≤ 𝐵, 𝐵, 0) < 𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
133	125	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
134		elioomnf 13347	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ if(0 ≤ 𝐵, 𝐵, 0) < 𝑦)))
135	133, 134	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ if(0 ≤ 𝐵, 𝐵, 0) < 𝑦)))
136		elioomnf 13347	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
137	133, 136	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
138	132, 135, 137	3bitr4d 311	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
139	96	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦)))
140	139	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦)))
141	43	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
142	141	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
143	138, 140, 142	3bitr4d 311	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦)))
144	143	pm5.32da 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → ((𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
145		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦))))
146	102, 103, 145	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦))))
147	146	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦))))
148		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
149	2, 53, 148	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
150	149	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
151	144, 147, 150	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
152	151	alrimiv 1927	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
153		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(-∞(,)𝑦)
154	111, 153	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦))
155	64, 153	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))
156	154, 155	cleqf 2920	. . . . 5 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
157	152, 156	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)))
158		mbfima 25529	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ∈ dom vol)
159	3, 102, 158	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ∈ dom vol)
160	159	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ∈ dom vol)
161	157, 160	eqeltrrd 2829	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
162		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 0)
163		simpllr 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
164	163	le0neg1d 11691	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
165	162, 164	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 0 ≤ -𝑦)
166	165	biantrurd 532	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝐵 ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
167	1	ad4ant14 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
168	163, 167	lenegd 11699	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
169		0red 11118	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
170	167	renegcld 11547	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
171	163	renegcld 11547	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
172		maxle 13093	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ ∧ -𝑦 ∈ ℝ) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
173	169, 170, 171, 172	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
174	166, 168, 173	3bitr4rd 312	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ 𝑦 ≤ 𝐵))
175	174	notbid 318	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (¬ if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ ¬ 𝑦 ≤ 𝐵))
176	23	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
177	171, 176	ltnled 11263	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝑦 < if(0 ≤ -𝐵, -𝐵, 0) ↔ ¬ if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦))
178	167, 163	ltnled 11263	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵))
179	175, 177, 178	3bitr4d 311	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝑦 < if(0 ≤ -𝐵, -𝐵, 0) ↔ 𝐵 < 𝑦))
180	176	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝑦 < if(0 ≤ -𝐵, -𝐵, 0) ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ -𝑦 < if(0 ≤ -𝐵, -𝐵, 0))))
181	167	biantrurd 532	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
182	179, 180, 181	3bitr3d 309	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ -𝑦 < if(0 ≤ -𝐵, -𝐵, 0)) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
183	171	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
184		elioopnf 13346	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ ℝ^* → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞) ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ -𝑦 < if(0 ≤ -𝐵, -𝐵, 0))))
185	183, 184	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞) ↔ (if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ ∧ -𝑦 < if(0 ≤ -𝐵, -𝐵, 0))))
186	163	rexrd 11165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
187	186, 136	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
188	182, 185, 187	3bitr4d 311	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞) ↔ 𝐵 ∈ (-∞(,)𝑦)))
189	38	eleq1d 2813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞)))
190	189	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞)))
191	141	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
192	188, 190, 191	3bitr4d 311	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦)))
193	192	pm5.32da 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → ((𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
194		elpreima 6992	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞))))
195	48, 49, 194	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞))))
196	195	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞))))
197	149	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
198	193, 196, 197	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
199	198	alrimiv 1927	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
200		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(-𝑦(,)+∞)
201	60, 200	nfima 6019	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞))
202	201, 155	cleqf 2920	. . . . 5 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
203	199, 202	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)))
204		mbfima 25529	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ∈ dom vol)
205	69, 48, 204	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ∈ dom vol)
206	205	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ∈ dom vol)
207	203, 206	eqeltrrd 2829	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
208	161, 207, 120, 119	ltlecasei 11224	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
209	2, 7, 121, 208	ismbf2d 25539	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2109 ifcif 4476 class class class wbr 5092 ↦ cmpt 5173 ^◡ccnv 5618 dom cdm 5619 “ cima 5622 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 +∞cpnf 11146 -∞cmnf 11147 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 -cneg 11348 (,)cioo 13248 volcvol 25362 MblFncmbf 25513

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

This theorem is referenced by: mbfposb 25552

W3C validator