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Theorem ismbfd 25702

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 25717. (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 13452	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6692	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3		ovelrn 7573	. . . . 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 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑥 ∈ ℝ^)
6		pnfxr 11237	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → +∞ ∈ ℝ^)
8		mnfxr 11240	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → -∞ ∈ ℝ^)
10		simprr 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑦 ∈ ℝ^)
11		iooin 13384	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (-∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
12	5, 7, 9, 10, 11	syl22anc 849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
13		ifcl 4527	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
14	8, 5, 13	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
15		mnfle 13138	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → -∞ ≤ 𝑥)
16		xrleid 13154	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
17		breq1 5104	. . . . . . . . . . . . . . 15 ⊢ (-∞ = if(𝑥 ≤ -∞, -∞, 𝑥) → (-∞ ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
18		breq1 5104	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = if(𝑥 ≤ -∞, -∞, 𝑥) → (𝑥 ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
19	17, 18	ifboth 4521	. . . . . . . . . . . . . 14 ⊢ ((-∞ ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
20	15, 16, 19	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ^* → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
21	20	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
22		xrmax1 13179	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
23	5, 8, 22	sylancl 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
24	14, 5, 21, 23	xrletrid 13158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) = 𝑥)
25		ifcl 4527	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
26	6, 10, 25	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
27		xrmin2 13182	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
28	6, 10, 27	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
29		pnfge 13133	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
30		xrleid 13154	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
31		breq2 5105	. . . . . . . . . . . . . . 15 ⊢ (+∞ = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ +∞ ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
32		breq2 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ 𝑦 ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
33	31, 32	ifboth 4521	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≤ +∞ ∧ 𝑦 ≤ 𝑦) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
34	29, 30, 33	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
35	34	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
36	26, 10, 28, 35	xrletrid 13158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) = 𝑦)
37	24, 36	oveq12d 7415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)) = (𝑥(,)𝑦))
38	12, 37	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (𝑥(,)𝑦))
39	38	imaeq2d 6050	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = (^◡𝐹 “ (𝑥(,)𝑦)))
40		ismbfd.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
41	40	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝐹:𝐴⟶ℝ)
42	41	ffund 6697	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → Fun 𝐹)
43		inpreima 7046	. . . . . . . . 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 2800	. . . . . . 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 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
48		ismbfd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
49	48	ralrimiva 3155	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
50		oveq2 7405	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (-∞(,)𝑥) = (-∞(,)𝑦))
51	50	imaeq2d 6050	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ (-∞(,)𝑦)))
52	51	eleq1d 2848	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol))
53	52	rspccva 3581	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
54	49, 53	sylan 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
55	54	adantrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
56		inmbl 25605	. . . . . . . 8 ⊢ (((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ∧ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
57	47, 55, 56	syl2anc 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
58	45, 57	eqeltrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol)
59		imaeq2 6046	. . . . . . 7 ⊢ (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝑥(,)𝑦)))
60	59	eleq1d 2848	. . . . . 6 ⊢ (𝑧 = (𝑥(,)𝑦) → ((^◡𝐹 “ 𝑧) ∈ dom vol ↔ (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol))
61	58, 60	syl5ibrcom 249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
62	61	rexlimdvva 3220	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
63	4, 62	biimtrid 244	. . 3 ⊢ (𝜑 → (𝑧 ∈ ran (,) → (^◡𝐹 “ 𝑧) ∈ dom vol))
64	63	ralrimiv 3154	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol)
65		ismbf 25691	. . 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 259	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1561 ∈ wcel 2143 ∀wral 3077 ∃wrex 3087 ∩ cin 3904 ifcif 4481 𝒫 cpw 4556 class class class wbr 5101 × cxp 5646 ^◡ccnv 5647 dom cdm 5648 ran crn 5649 “ cima 5651 Fun wfun 6516 Fn wfn 6517 ⟶wf 6518 (class class class)co 7397 ℝcr 11073 +∞cpnf 11214 -∞cmnf 11215 ℝ^*cxr 11216 ≤ cle 11218 (,)cioo 13350 volcvol 25526 MblFncmbf 25677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-inf2 9597 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-of 7661 df-om 7848 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-er 8679 df-map 8811 df-pm 8812 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-sup 9389 df-inf 9390 df-oi 9459 df-dju 9860 df-card 9898 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-n0 12483 df-z 12570 df-uz 12841 df-q 12951 df-rp 12995 df-xadd 13116 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13514 df-fzo 13661 df-fl 13803 df-seq 14016 df-exp 14076 df-hash 14345 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-clim 15516 df-sum 15715 df-xmet 21418 df-met 21419 df-ovol 25527 df-vol 25528 df-mbf 25682

This theorem is referenced by: ismbf2d 25703 mbfmax 25712

W3C validator