orvclteinc - Mathbox for Thierry Arnoux

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Theorem orvclteinc 35091

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 35063	. . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ)
4	3	ffund 6706	. . 3 ⊢ (𝜑 → Fun 𝑋)
5		simp2 1155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ)
6		orvclteinc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7	6	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ)
8		orvclteinc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ)
10		simp3 1156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴)
11		orvclteinc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	11	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵)
13	5, 7, 9, 10, 12	letrd 11448	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵)
14	13	3expia 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵))
15	14	ss2rabdv 4023	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})
16		sspreima 7059	. . 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 596	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
18	1, 2, 6	orrvcval4 35080	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
19	1, 2, 8	orrvcval4 35080	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐵) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
20	17, 18, 19	3sstr4d 3986	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 ∈ wcel 2145 {crab 3413 ⊆ wss 3899 class class class wbr 5103 ^◡ccnv 5650 dom cdm 5651 “ cima 5654 Fun wfun 6525 ‘cfv 6531 (class class class)co 7412 ℝcr 11180 ≤ cle 11325 Probcprb 35022 rRndVarcrrv 35055 ∘_RV/𝑐corvc 35071

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-pre-lttri 11255 ax-pre-lttrn 11256

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-ioo 13461 df-topgen 17594 df-top 23192 df-bases 23244 df-esum 34642 df-siga 34723 df-sigagen 34754 df-brsiga 34797 df-meas 34811 df-mbfm 34865 df-prob 35023 df-rrv 35056 df-orvc 35072

This theorem is used by: dstfrvinc 35092 dstfrvclim1 35093

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