dstfrvel - Mathbox for Thierry Arnoux

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Theorem dstfrvel 34455

Description: Elementhood of preimage maps produced by the "less than or equal to" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)

**Hypotheses**
Ref	Expression
dstfrv.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstfrv.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orvclteel.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
dstfrvel.1	⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃)
dstfrvel.2	⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴)

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

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

Step	Hyp	Ref	Expression
1		dstfrv.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		dstfrv.2	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
3	1, 2	rrvvf 34426	. . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
4		dstfrvel.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃)
5	3, 4	ffvelcdmd 7105	. . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ)
6		dstfrvel.2	. . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴)
7		breq1 5151	. . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴))
8	7	elrab 3695	. . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴))
9	5, 6, 8	sylanbrc 583	. . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})
10	3	ffund 6741	. . . 4 ⊢ (𝜑 → Fun 𝑋)
11	1, 2	rrvdm 34428	. . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃)
12	4, 11	eleqtrrd 2842	. . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋)
13		fvimacnv 7073	. . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})))
14	10, 12, 13	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})))
15	9, 14	mpbid 232	. 2 ⊢ (𝜑 → 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
16		orvclteel.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
17	1, 2, 16	orrvcval4 34446	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
18	15, 17	eleqtrrd 2842	1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘_RV/𝑐 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 {crab 3433 ∪ cuni 4912 class class class wbr 5148 ^◡ccnv 5688 dom cdm 5689 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 ≤ cle 11294 Probcprb 34389 rRndVarcrrv 34422 ∘_RV/𝑐corvc 34437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 df-topgen 17490 df-top 22916 df-bases 22969 df-esum 34009 df-siga 34090 df-sigagen 34120 df-brsiga 34163 df-meas 34177 df-mbfm 34231 df-prob 34390 df-rrv 34423 df-orvc 34438

This theorem is referenced by: dstfrvunirn 34456

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