orvclteinc - Mathbox for Thierry Arnoux

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Theorem orvclteinc 34653

Description: Preimage maps produced by the "less than or equal to" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)

**Hypotheses**
Ref	Expression
dstfrv.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstfrv.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orvclteinc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
orvclteinc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
orvclteinc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
orvclteinc	⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Proof of Theorem orvclteinc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dstfrv.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		dstfrv.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
3	1, 2	rrvf2 34625	. . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ)
4	3	ffund 6674	. . 3 ⊢ (𝜑 → Fun 𝑋)
5		simp2 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ)
6		orvclteinc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7	6	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ)
8		orvclteinc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ)
10		simp3 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴)
11		orvclteinc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	11	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵)
13	5, 7, 9, 10, 12	letrd 11302	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵)
14	13	3expia 1122	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵))
15	14	ss2rabdv 4029	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})
16		sspreima 7022	. . 3 ⊢ ((Fun 𝑋 ∧ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}) → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
17	4, 15, 16	syl2anc 585	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
18	1, 2, 6	orrvcval4 34642	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
19	1, 2, 8	orrvcval4 34642	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐵) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
20	17, 18, 19	3sstr4d 3991	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2114 {crab 3401 ⊆ wss 3903 class class class wbr 5100 ^◡ccnv 5631 dom cdm 5632 “ cima 5635 Fun wfun 6494 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 ≤ cle 11179 Probcprb 34584 rRndVarcrrv 34617 ∘_RV/𝑐corvc 34633

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-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

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-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-ioo 13277 df-topgen 17375 df-top 22850 df-bases 22902 df-esum 34205 df-siga 34286 df-sigagen 34316 df-brsiga 34359 df-meas 34373 df-mbfm 34427 df-prob 34585 df-rrv 34618 df-orvc 34634

This theorem is referenced by: dstfrvinc 34654 dstfrvclim1 34655

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