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Theorem ismbfd 25619

Description: Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 25634. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Hypotheses**
Ref	Expression
ismbfd.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
ismbfd.2	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
ismbfd.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)

**Assertion**
Ref	Expression
ismbfd	⊢ (𝜑 → 𝐹 ∈ MblFn)

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

Proof of Theorem ismbfd Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioof 13394	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6663	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3		ovelrn 7537	. . . . 5 ⊢ ((,) Fn (ℝ^* × ℝ^) → (𝑧 ∈ ran (,) ↔ ∃𝑥 ∈ ℝ^ ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦)))
4	1, 2, 3	mp2b 10	. . . 4 ⊢ (𝑧 ∈ ran (,) ↔ ∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦))
5		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑥 ∈ ℝ^)
6		pnfxr 11193	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → +∞ ∈ ℝ^)
8		mnfxr 11196	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → -∞ ∈ ℝ^)
10		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑦 ∈ ℝ^)
11		iooin 13326	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (-∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
12	5, 7, 9, 10, 11	syl22anc 839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
13		ifcl 4513	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
14	8, 5, 13	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
15		mnfle 13080	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → -∞ ≤ 𝑥)
16		xrleid 13096	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
17		breq1 5089	. . . . . . . . . . . . . . 15 ⊢ (-∞ = if(𝑥 ≤ -∞, -∞, 𝑥) → (-∞ ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
18		breq1 5089	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = if(𝑥 ≤ -∞, -∞, 𝑥) → (𝑥 ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
19	17, 18	ifboth 4507	. . . . . . . . . . . . . 14 ⊢ ((-∞ ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
20	15, 16, 19	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ^* → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
21	20	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
22		xrmax1 13121	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
23	5, 8, 22	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
24	14, 5, 21, 23	xrletrid 13100	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) = 𝑥)
25		ifcl 4513	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
26	6, 10, 25	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
27		xrmin2 13124	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
28	6, 10, 27	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
29		pnfge 13075	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
30		xrleid 13096	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
31		breq2 5090	. . . . . . . . . . . . . . 15 ⊢ (+∞ = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ +∞ ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
32		breq2 5090	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ 𝑦 ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
33	31, 32	ifboth 4507	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≤ +∞ ∧ 𝑦 ≤ 𝑦) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
34	29, 30, 33	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
35	34	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
36	26, 10, 28, 35	xrletrid 13100	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) = 𝑦)
37	24, 36	oveq12d 7379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)) = (𝑥(,)𝑦))
38	12, 37	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (𝑥(,)𝑦))
39	38	imaeq2d 6020	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = (^◡𝐹 “ (𝑥(,)𝑦)))
40		ismbfd.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝐹:𝐴⟶ℝ)
42	41	ffund 6667	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → Fun 𝐹)
43		inpreima 7011	. . . . . . . . 9 ⊢ (Fun 𝐹 → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
45	39, 44	eqtr3d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
46		ismbfd.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
47	46	adantrr 718	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
48		ismbfd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
49	48	ralrimiva 3130	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
50		oveq2 7369	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (-∞(,)𝑥) = (-∞(,)𝑦))
51	50	imaeq2d 6020	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ (-∞(,)𝑦)))
52	51	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol))
53	52	rspccva 3564	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
54	49, 53	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
55	54	adantrl 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
56		inmbl 25522	. . . . . . . 8 ⊢ (((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ∧ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
57	47, 55, 56	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
58	45, 57	eqeltrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol)
59		imaeq2 6016	. . . . . . 7 ⊢ (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝑥(,)𝑦)))
60	59	eleq1d 2822	. . . . . 6 ⊢ (𝑧 = (𝑥(,)𝑦) → ((^◡𝐹 “ 𝑧) ∈ dom vol ↔ (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol))
61	58, 60	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
62	61	rexlimdvva 3195	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
63	4, 62	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑧 ∈ ran (,) → (^◡𝐹 “ 𝑧) ∈ dom vol))
64	63	ralrimiv 3129	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol)
65		ismbf 25608	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol))
66	40, 65	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol))
67	64, 66	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∩ cin 3889 ifcif 4467 𝒫 cpw 4542 class class class wbr 5086 × cxp 5623 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 (class class class)co 7361 ℝcr 11031 +∞cpnf 11170 -∞cmnf 11171 ℝ^*cxr 11172 ≤ cle 11174 (,)cioo 13292 volcvol 25443 MblFncmbf 25594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-z 12519 df-uz 12783 df-q 12893 df-rp 12937 df-xadd 13058 df-ioo 13296 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-sum 15643 df-xmet 21340 df-met 21341 df-ovol 25444 df-vol 25445 df-mbf 25599

This theorem is referenced by: ismbf2d 25620 mbfmax 25629

W3C validator