dstfrvel - Mathbox for Thierry Arnoux

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Theorem dstfrvel 34665

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 34635	. . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
4		dstfrvel.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃)
5	3, 4	ffvelcdmd 7033	. . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ)
6		dstfrvel.2	. . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴)
7		breq1 5082	. . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴))
8	7	elrab 3636	. . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴))
9	5, 6, 8	sylanbrc 589	. . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})
10	3	ffund 6666	. . . 4 ⊢ (𝜑 → Fun 𝑋)
11	1, 2	rrvdm 34637	. . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃)
12	4, 11	eleqtrrd 2843	. . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋)
13		fvimacnv 7001	. . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})))
14	10, 12, 13	syl2anc 590	. . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})))
15	9, 14	mpbid 233	. 2 ⊢ (𝜑 → 𝐵 ∈ (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
16		orvclteel.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
17	1, 2, 16	orrvcval4 34656	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) = (^◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))
18	15, 17	eleqtrrd 2843	1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘_RV/𝑐 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2119 {crab 3392 ∪ cuni 4845 class class class wbr 5079 ^◡ccnv 5624 dom cdm 5625 “ cima 5628 Fun wfun 6486 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 ≤ cle 11178 Probcprb 34598 rRndVarcrrv 34631 ∘_RV/𝑐corvc 34647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-ioo 13300 df-topgen 17404 df-top 22884 df-bases 22936 df-esum 34219 df-siga 34300 df-sigagen 34330 df-brsiga 34373 df-meas 34387 df-mbfm 34441 df-prob 34599 df-rrv 34632 df-orvc 34648

This theorem is referenced by: dstfrvunirn 34666

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