orrvcval4 - Mathbox for Thierry Arnoux

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Theorem orrvcval4 34429

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 34376	. . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
4		retop 24803	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
5	4	a1i 11	. . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
6		orrvccel.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
7	1	rrvmbfm 34407	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
8	6, 7	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
9		df-brsiga 34146	. . . . 5 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
10	9	oveq2i 7459	. . . 4 ⊢ (dom 𝑃MblFnM𝔅_ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))
11	8, 10	eleqtrdi 2854	. . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))))
12		orrvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	3, 5, 11, 12	orvcval4 34425	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}))
14		uniretop 24804	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
15		rabeq 3458	. . . 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 6087	. 2 ⊢ (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})
18	13, 17	eqtr4di 2798	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 {crab 3443 ∪ cuni 4931 class class class wbr 5166 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 “ cima 5703 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 (,)cioo 13407 topGenctg 17497 Topctop 22920 sigAlgebracsiga 34072 sigaGencsigagen 34102 𝔅_ℝcbrsiga 34145 MblFnMcmbfm 34213 Probcprb 34372 rRndVarcrrv 34405 ∘_RV/𝑐corvc 34420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-ioo 13411 df-topgen 17503 df-top 22921 df-bases 22974 df-esum 33992 df-siga 34073 df-sigagen 34103 df-brsiga 34146 df-meas 34160 df-mbfm 34214 df-prob 34373 df-rrv 34406 df-orvc 34421

This theorem is referenced by: orvcelval 34433 dstfrvel 34438 orvclteinc 34440

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