orrvcval4 - Mathbox for Thierry Arnoux

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Theorem orrvcval4 34433

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 34379	. . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
4		retop 24647	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
5	4	a1i 11	. . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
6		orrvccel.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
7	1	rrvmbfm 34410	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
8	6, 7	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
9		df-brsiga 34149	. . . . 5 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
10	9	oveq2i 7360	. . . 4 ⊢ (dom 𝑃MblFnM𝔅_ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))
11	8, 10	eleqtrdi 2838	. . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))))
12		orrvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13	3, 5, 11, 12	orvcval4 34429	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}))
14		uniretop 24648	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
15		rabeq 3409	. . . 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 6009	. 2 ⊢ (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})
18	13, 17	eqtr4di 2782	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 {crab 3394 ∪ cuni 4858 class class class wbr 5092 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 “ cima 5622 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 (,)cioo 13248 topGenctg 17341 Topctop 22778 sigAlgebracsiga 34075 sigaGencsigagen 34105 𝔅_ℝcbrsiga 34148 MblFnMcmbfm 34216 Probcprb 34375 rRndVarcrrv 34408 ∘_RV/𝑐corvc 34424

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-ioo 13252 df-topgen 17347 df-top 22779 df-bases 22831 df-esum 33995 df-siga 34076 df-sigagen 34106 df-brsiga 34149 df-meas 34163 df-mbfm 34217 df-prob 34376 df-rrv 34409 df-orvc 34425

This theorem is referenced by: orvcelval 34437 dstfrvel 34442 orvclteinc 34444

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