orrvcval4 - Mathbox for Thierry Arnoux

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Theorem orrvcval4 34497

Description: The value of the preimage mapping operator can be restricted to preimages of subsets of ℝ. (Contributed by Thierry Arnoux, 21-Jan-2017.)

**Hypotheses**
Ref	Expression
orrvccel.1	⊢ (𝜑 → 𝑃 ∈ Prob)
orrvccel.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orrvccel.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)

**Assertion**
Ref	Expression
orrvcval4	⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Distinct variable groups: 𝑦,𝐴 𝑦,𝑅 𝑦,𝑋 Allowed substitution hints: 𝜑(𝑦) 𝑃(𝑦) 𝑉(𝑦)

**Proof of Theorem orrvcval4**
Step	Hyp	Ref	Expression
1		orrvccel.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		domprobsiga 34443	. . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
4		retop 24700	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
5	4	a1i 11	. . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
6		orrvccel.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
7	1	rrvmbfm 34474	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
8	6, 7	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
9		df-brsiga 34213	. . . . 5 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
10	9	oveq2i 7416	. . . 4 ⊢ (dom 𝑃MblFnM𝔅_ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))
11	8, 10	eleqtrdi 2844	. . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))))
12		orrvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	3, 5, 11, 12	orvcval4 34493	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}))
14		uniretop 24701	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
15		rabeq 3430	. . . 4 ⊢ (ℝ = ∪ (topGen‘ran (,)) → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})
16	14, 15	ax-mp 5	. . 3 ⊢ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}
17	16	imaeq2i 6045	. 2 ⊢ (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})
18	13, 17	eqtr4di 2788	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3415 ∪ cuni 4883 class class class wbr 5119 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 “ cima 5657 ‘cfv 6531 (class class class)co 7405 ℝcr 11128 (,)cioo 13362 topGenctg 17451 Topctop 22831 sigAlgebracsiga 34139 sigaGencsigagen 34169 𝔅_ℝcbrsiga 34212 MblFnMcmbfm 34280 Probcprb 34439 rRndVarcrrv 34472 ∘_RV/𝑐corvc 34488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-ioo 13366 df-topgen 17457 df-top 22832 df-bases 22884 df-esum 34059 df-siga 34140 df-sigagen 34170 df-brsiga 34213 df-meas 34227 df-mbfm 34281 df-prob 34440 df-rrv 34473 df-orvc 34489

This theorem is referenced by: orvcelval 34501 dstfrvel 34506 orvclteinc 34508

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