incsmflem - Mathbox for Glauco Siliprandi

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Theorem incsmflem 44914

Description: A nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
incsmflem.x	⊢ Ⅎ𝑥𝜑
incsmflem.y	⊢ Ⅎ𝑦𝜑
incsmflem.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
incsmflem.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
incsmflem.i	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
incsmflem.j	⊢ 𝐽 = (topGen‘ran (,))
incsmflem.b	⊢ 𝐵 = (SalGen‘𝐽)
incsmflem.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
incsmflem.l	⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
incsmflem.c	⊢ 𝐶 = sup(𝑌, ℝ^*, < )
incsmflem.d	⊢ 𝐷 = (-∞(,)𝐶)
incsmflem.e	⊢ 𝐸 = (-∞(,]𝐶)

**Assertion**
Ref	Expression
incsmflem	⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))

Distinct variable groups: 𝐴,𝑏 𝑥,𝐴,𝑦 𝐵,𝑏 𝑥,𝐶,𝑦 𝐷,𝑏 𝑥,𝐷,𝑦 𝐸,𝑏 𝑥,𝐸,𝑦 𝑥,𝐹,𝑦 𝑥,𝑅,𝑦 𝑌,𝑏 𝑦,𝑌 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑏) 𝐵(𝑥,𝑦) 𝐶(𝑏) 𝑅(𝑏) 𝐹(𝑏) 𝐽(𝑥,𝑦,𝑏) 𝑌(𝑥)

**Proof of Theorem incsmflem**
Step	Hyp	Ref	Expression
1		incsmflem.e	. . . 4 ⊢ 𝐸 = (-∞(,]𝐶)
2		mnfxr 11208	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		incsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
5		ssrab2 4035	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴
6	4, 5	eqsstri 3976	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		incsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3952	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3942	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ ℝ)
11		incsmflem.j	. . . . 5 ⊢ 𝐽 = (topGen‘ran (,))
12		incsmflem.b	. . . . 5 ⊢ 𝐵 = (SalGen‘𝐽)
13	3, 10, 11, 12	iocborel 44529	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2842	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		incsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑥𝐶
17		nfrab1 3424	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
18	4, 17	nfcxfr 2903	. . . . . 6 ⊢ Ⅎ𝑥𝑌
19	16, 18	nfel 2919	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌
20	15, 19	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌)
21		incsmflem.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
22		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌
23	21, 22	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐶 ∈ 𝑌)
24	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
25		incsmflem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
27		incsmflem.i	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
28	27	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
29		incsmflem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
30	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
31		incsmflem.c	. . . 4 ⊢ 𝐶 = sup(𝑌, ℝ^*, < )
32		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌)
33	20, 23, 24, 26, 28, 30, 4, 31, 32, 1	pimincfltioc 44889	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4163	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3593	. . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
36	14, 33, 35	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
37		incsmflem.d	. . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶)
38	11, 12	iooborel 44524	. . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵
39	37, 38	eqeltri 2834	. . . . 5 ⊢ 𝐷 ∈ 𝐵
40	39	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
41	40	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵)
42	19	nfn 1860	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌
43	15, 42	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
44		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌
45	21, 44	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
46	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
47	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
48	27	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
49	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
50		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌)
51	43, 45, 46, 47, 48, 49, 4, 31, 50, 37	pimincfltioo 44891	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
52		ineq1 4163	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
53	52	rspceeqv 3593	. . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
54	41, 51, 53	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
55	36, 54	pm2.61dan 811	1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 {crab 3405 ∩ cin 3907 ⊆ wss 3908 class class class wbr 5103 ran crn 5632 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 supcsup 9372 ℝcr 11046 -∞cmnf 11183 ℝ^*cxr 11184 < clt 11185 ≤ cle 11186 (,)cioo 13256 (,]cioc 13257 topGenctg 17311 SalGencsalgen 44485

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 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 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-card 9871 df-acn 9874 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-q 12866 df-rp 12908 df-ioo 13260 df-ioc 13261 df-fl 13689 df-topgen 17317 df-top 22227 df-bases 22280 df-salg 44482 df-salgen 44486

This theorem is referenced by: incsmf 44915

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