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Theorem mbfposr 25880

Description: Converse to mbfpos 25879. (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 7108	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		mbfposr.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
4		0re 11234	. . . 4 ⊢ 0 ∈ ℝ
5		ifcl 4528	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
6	1, 4, 5	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
7	3, 6	mbfdm2 25865	. 2 ⊢ (𝜑 → 𝐴 ∈ dom vol)
8		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 < 0)
9		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
10	9	lt0neg1d 11807	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 0 ↔ 0 < -𝑦))
11	8, 10	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 0 < -𝑦)
12	11	biantrurd 542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (-𝐵 < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
13	1	ad4ant14 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	9, 13	ltnegd 11816	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ -𝐵 < -𝑦))
15		0red 11235	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
16	13	renegcld 11665	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
17	9	renegcld 11665	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
18		maxlt 13245	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ ∧ -𝑦 ∈ ℝ) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
19	15, 16, 17, 18	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ (0 < -𝑦 ∧ -𝐵 < -𝑦)))
20	12, 14, 19	3bitr4rd 315	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) < -𝑦 ↔ 𝑦 < 𝐵))
21	1	renegcld 11665	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
22		ifcl 4528	. . . . . . . . . . . . . 14 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
23	21, 4, 22	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
24	23	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
25	24	biantrurd 542	. . . . . . . . . . 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 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
27	20, 25, 26	3bitr3d 312	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
29		elioomnf 13497	. . . . . . . . . . 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 18	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
32		elioopnf 13496	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
33	31, 32	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
34	27, 30, 33	3bitr4d 314	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦) ↔ 𝐵 ∈ (𝑦(,)+∞)))
35		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
36		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
37	36	fvmpt2 6998	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) = if(0 ≤ -𝐵, -𝐵, 0))
39	38	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-∞(,)-𝑦) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-∞(,)-𝑦)))
40	39	ad4ant14 765	. . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
42	41	fvmpt2 6998	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
43	35, 1, 42	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
44	43	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
45	44	ad4ant14 765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
46	34, 40, 45	3bitr4d 314	. . . . . . . 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 590	. . . . . . 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 7108	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ)
49		ffn 6702	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) Fn 𝐴)
50		elpreima 7050	. . . . . . . . 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 19	. . . . . . . 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 739	. . . . . . 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 6702	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
54		elpreima 7050	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
55	2, 53, 54	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
56	55	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
57	47, 52, 56	3bitr4d 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
58	57	alrimiv 1960	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
59		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
60	59	nfcnv 5858	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))
61		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥(-∞(,)-𝑦)
62	60, 61	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦))
63		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
64	63	nfcnv 5858	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ 𝐵)
65		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦(,)+∞)
66	64, 65	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))
67	62, 66	cleqf 2950	. . . . 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 237	. . . 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 25858	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ∈ dom vol)
72	71	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-∞(,)-𝑦)) ∈ dom vol)
73	68, 72	eqeltrrd 2861	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 0) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
74		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝑦)
75		0red 11235	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
76	1	ad4ant14 765	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
77		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
78		maxle 13243	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ (0 ≤ 𝑦 ∧ 𝐵 ≤ 𝑦)))
79	75, 76, 77, 78	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ (0 ≤ 𝑦 ∧ 𝐵 ≤ 𝑦)))
80	74, 79	mpbirand 720	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
81	80	notbid 321	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (¬ if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦 ↔ ¬ 𝐵 ≤ 𝑦))
82	76, 4, 5	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
83	77, 82	ltnled 11381	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < if(0 ≤ 𝐵, 𝐵, 0) ↔ ¬ if(0 ≤ 𝐵, 𝐵, 0) ≤ 𝑦))
84	77, 76	ltnled 11381	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ ¬ 𝐵 ≤ 𝑦))
85	81, 83, 84	3bitr4d 314	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑦 < 𝐵))
86	82	biantrurd 542	. . . . . . . . . . 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 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑦 < 𝐵 ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
88	85, 86, 87	3bitr3d 312	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
90		elioopnf 13496	. . . . . . . . . . 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 18	. . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (𝑦(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑦 < 𝐵)))
93	88, 91, 92	3bitr4d 314	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
94		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
95	94	fvmpt2 6998	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) = if(0 ≤ 𝐵, 𝐵, 0))
97	96	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (𝑦(,)+∞) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (𝑦(,)+∞)))
98	97	ad4ant14 765	. . . . . . . . 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 765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞) ↔ 𝐵 ∈ (𝑦(,)+∞)))
100	93, 98, 99	3bitr4d 314	. . . . . . . 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 590	. . . . . . 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 7108	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ)
103		ffn 6702	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℝ → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) Fn 𝐴)
104		elpreima 7050	. . . . . . . . 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 19	. . . . . . . 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 739	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑦(,)+∞))))
108	101, 106, 107	3bitr4d 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
109	108	alrimiv 1960	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞))))
110		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
111	110	nfcnv 5858	. . . . . . 7 ⊢ Ⅎ𝑥^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))
112	111, 65	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞))
113	112, 66	cleqf 2950	. . . . 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 237	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)))
115		mbfima 25858	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ∈ dom vol)
117	116	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (𝑦(,)+∞)) ∈ dom vol)
118	114, 117	eqeltrrd 2861	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 ≤ 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
119		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
120		0red 11235	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℝ)
121	73, 118, 119, 120	ltlecasei 11342	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑦(,)+∞)) ∈ dom vol)
122		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 < 𝑦)
123		0red 11235	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
124	1	ad4ant14 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
125		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
126		maxlt 13245	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ (0 < 𝑦 ∧ 𝐵 < 𝑦)))
127	123, 124, 125, 126	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ (0 < 𝑦 ∧ 𝐵 < 𝑦)))
128	122, 127	mpbirand 720	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) < 𝑦 ↔ 𝐵 < 𝑦))
129	6	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
130	129	biantrurd 542	. . . . . . . . . . 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 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
132	128, 130, 131	3bitr3d 312	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
134		elioomnf 13497	. . . . . . . . . . 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 18	. . . . . . . . . 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 13497	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
137	133, 136	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
138	132, 135, 137	3bitr4d 314	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
139	96	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))‘𝑥) ∈ (-∞(,)𝑦) ↔ if(0 ≤ 𝐵, 𝐵, 0) ∈ (-∞(,)𝑦)))
140	139	ad4ant14 765	. . . . . . . . 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 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
142	141	ad4ant14 765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
143	138, 140, 142	3bitr4d 314	. . . . . . . 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 590	. . . . . . 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 7050	. . . . . . . . 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 19	. . . . . . . 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 739	. . . . . . 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 7050	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
149	2, 53, 148	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
150	149	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
151	144, 147, 150	3bitr4d 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
152	151	alrimiv 1960	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
153		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥(-∞(,)𝑦)
154	111, 153	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦))
155	64, 153	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))
156	154, 155	cleqf 2950	. . . . 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 237	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)))
158		mbfima 25858	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ∈ dom vol)
160	159	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) “ (-∞(,)𝑦)) ∈ dom vol)
161	157, 160	eqeltrrd 2861	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 0 < 𝑦) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
162		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 0)
163		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
164	163	le0neg1d 11809	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
165	162, 164	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 0 ≤ -𝑦)
166	165	biantrurd 542	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝐵 ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
167	1	ad4ant14 765	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
168	163, 167	lenegd 11817	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
169		0red 11235	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
170	167	renegcld 11665	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
171	163	renegcld 11665	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
172		maxle 13243	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ ∧ -𝑦 ∈ ℝ) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
173	169, 170, 171, 172	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ (0 ≤ -𝑦 ∧ -𝐵 ≤ -𝑦)))
174	166, 168, 173	3bitr4rd 315	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ 𝑦 ≤ 𝐵))
175	174	notbid 321	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (¬ if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦 ↔ ¬ 𝑦 ≤ 𝐵))
176	23	ad4ant14 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
177	171, 176	ltnled 11381	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝑦 < if(0 ≤ -𝐵, -𝐵, 0) ↔ ¬ if(0 ≤ -𝐵, -𝐵, 0) ≤ -𝑦))
178	167, 163	ltnled 11381	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵))
179	175, 177, 178	3bitr4d 314	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (-𝑦 < if(0 ≤ -𝐵, -𝐵, 0) ↔ 𝐵 < 𝑦))
180	176	biantrurd 542	. . . . . . . . . . 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 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 < 𝑦 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
182	179, 180, 181	3bitr3d 312	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
184		elioopnf 13496	. . . . . . . . . . 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 18	. . . . . . . . . 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 11283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
187	186, 136	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)𝑦) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝑦)))
188	182, 185, 187	3bitr4d 314	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞) ↔ 𝐵 ∈ (-∞(,)𝑦)))
189	38	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))‘𝑥) ∈ (-𝑦(,)+∞) ↔ if(0 ≤ -𝐵, -𝐵, 0) ∈ (-𝑦(,)+∞)))
190	189	ad4ant14 765	. . . . . . . . 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 765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦) ↔ 𝐵 ∈ (-∞(,)𝑦)))
192	188, 190, 191	3bitr4d 314	. . . . . . . 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 590	. . . . . . 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 7050	. . . . . . . . 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 19	. . . . . . . 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 739	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (-∞(,)𝑦))))
198	193, 196, 197	3bitr4d 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
199	198	alrimiv 1960	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → ∀𝑥(𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ↔ 𝑥 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦))))
200		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥(-𝑦(,)+∞)
201	60, 200	nfima 6064	. . . . . 6 ⊢ Ⅎ𝑥(^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞))
202	201, 155	cleqf 2950	. . . . 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 237	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)))
204		mbfima 25858	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ∈ dom vol)
206	205	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) “ (-𝑦(,)+∞)) ∈ dom vol)
207	203, 206	eqeltrrd 2861	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑦 ≤ 0) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
208	161, 207, 120, 119	ltlecasei 11342	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)𝑦)) ∈ dom vol)
209	2, 7, 121, 208	ismbf2d 25868	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2145 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5654 dom cdm 5655 “ cima 5658 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℝcr 11123 0cc0 11124 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 -cneg 11466 (,)cioo 13398 volcvol 25691 MblFncmbf 25842

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

This theorem is used by: mbfposb 25881

W3C validator