orvclteinc - Mathbox for Thierry Arnoux

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Theorem orvclteinc 34633

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 34605	. . . 4 ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ)
4	3	ffund 6666	. . 3 ⊢ (𝜑 → Fun 𝑋)
5		simp2 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ∈ ℝ)
6		orvclteinc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7	6	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ∈ ℝ)
8		orvclteinc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐵 ∈ ℝ)
10		simp3 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐴)
11		orvclteinc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	11	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝐴 ≤ 𝐵)
13	5, 7, 9, 10, 12	letrd 11290	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) → 𝑥 ≤ 𝐵)
14	13	3expia 1121	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 → 𝑥 ≤ 𝐵))
15	14	ss2rabdv 4027	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ⊆ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵})
16		sspreima 7013	. . 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 584	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) ⊆ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
18	1, 2, 6	orrvcval4 34622	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
19	1, 2, 8	orrvcval4 34622	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐵) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐵}))
20	17, 18, 19	3sstr4d 3989	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘_RV/𝑐 ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2113 {crab 3399 ⊆ wss 3901 class class class wbr 5098 ^◡ccnv 5623 dom cdm 5624 “ cima 5627 Fun wfun 6486 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 ≤ cle 11167 Probcprb 34564 rRndVarcrrv 34597 ∘_RV/𝑐corvc 34613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-pre-lttri 11100 ax-pre-lttrn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-ioo 13265 df-topgen 17363 df-top 22838 df-bases 22890 df-esum 34185 df-siga 34266 df-sigagen 34296 df-brsiga 34339 df-meas 34353 df-mbfm 34407 df-prob 34565 df-rrv 34598 df-orvc 34614

This theorem is referenced by: dstfrvinc 34634 dstfrvclim1 34635

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