orrvcval4 - Mathbox for Thierry Arnoux

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Theorem orrvcval4 34467

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 34413	. . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
4		retop 24782	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
5	4	a1i 11	. . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
6		orrvccel.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
7	1	rrvmbfm 34444	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
8	6, 7	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
9		df-brsiga 34183	. . . . 5 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
10	9	oveq2i 7442	. . . 4 ⊢ (dom 𝑃MblFnM𝔅_ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))
11	8, 10	eleqtrdi 2851	. . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))))
12		orrvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	3, 5, 11, 12	orvcval4 34463	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}))
14		uniretop 24783	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
15		rabeq 3451	. . . 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 6076	. 2 ⊢ (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})
18	13, 17	eqtr4di 2795	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ∪ cuni 4907 class class class wbr 5143 ^◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 (,)cioo 13387 topGenctg 17482 Topctop 22899 sigAlgebracsiga 34109 sigaGencsigagen 34139 𝔅_ℝcbrsiga 34182 MblFnMcmbfm 34250 Probcprb 34409 rRndVarcrrv 34442 ∘_RV/𝑐corvc 34458

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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-bases 22953 df-esum 34029 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 df-orvc 34459

This theorem is referenced by: orvcelval 34471 dstfrvel 34476 orvclteinc 34478

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