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Theorem ismbfd 25606

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 25621. (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 13400	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6668	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3		ovelrn 7543	. . . . 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 11199	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → +∞ ∈ ℝ^)
8		mnfxr 11202	. . . . . . . . . . . 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 13332	. . . . . . . . . . 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 4512	. . . . . . . . . . . . 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 13086	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → -∞ ≤ 𝑥)
16		xrleid 13102	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
17		breq1 5088	. . . . . . . . . . . . . . 15 ⊢ (-∞ = if(𝑥 ≤ -∞, -∞, 𝑥) → (-∞ ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
18		breq1 5088	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = if(𝑥 ≤ -∞, -∞, 𝑥) → (𝑥 ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
19	17, 18	ifboth 4506	. . . . . . . . . . . . . 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 13127	. . . . . . . . . . . . 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 13106	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) = 𝑥)
25		ifcl 4512	. . . . . . . . . . . . 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 13130	. . . . . . . . . . . . 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 13081	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
30		xrleid 13102	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
31		breq2 5089	. . . . . . . . . . . . . . 15 ⊢ (+∞ = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ +∞ ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
32		breq2 5089	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ 𝑦 ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
33	31, 32	ifboth 4506	. . . . . . . . . . . . . 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 13106	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) = 𝑦)
37	24, 36	oveq12d 7385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)) = (𝑥(,)𝑦))
38	12, 37	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (𝑥(,)𝑦))
39	38	imaeq2d 6025	. . . . . . . 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 6672	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → Fun 𝐹)
43		inpreima 7016	. . . . . . . . 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 2773	. . . . . . 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 3129	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
50		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (-∞(,)𝑥) = (-∞(,)𝑦))
51	50	imaeq2d 6025	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ (-∞(,)𝑦)))
52	51	eleq1d 2821	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol))
53	52	rspccva 3563	. . . . . . . . . 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 25509	. . . . . . . 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 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol)
59		imaeq2 6021	. . . . . . 7 ⊢ (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝑥(,)𝑦)))
60	59	eleq1d 2821	. . . . . 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 3194	. . . 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 3128	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol)
65		ismbf 25595	. . 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 3051 ∃wrex 3061 ∩ cin 3888 ifcif 4466 𝒫 cpw 4541 class class class wbr 5085 × cxp 5629 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fun wfun 6492 Fn wfn 6493 ⟶wf 6494 (class class class)co 7367 ℝcr 11037 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 ≤ cle 11180 (,)cioo 13298 volcvol 25430 MblFncmbf 25581

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xadd 13064 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-xmet 21345 df-met 21346 df-ovol 25431 df-vol 25432 df-mbf 25586

This theorem is referenced by: ismbf2d 25607 mbfmax 25616

W3C validator