orrvcval4 - Mathbox for Thierry Arnoux

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Theorem orrvcval4 34449

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 34395	. . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
4		retop 24682	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
5	4	a1i 11	. . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
6		orrvccel.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
7	1	rrvmbfm 34426	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
8	6, 7	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
9		df-brsiga 34165	. . . . 5 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
10	9	oveq2i 7380	. . . 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 34445	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}))
14		uniretop 24683	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
15		rabeq 3417	. . . 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 6018	. 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 3402 ∪ cuni 4867 class class class wbr 5102 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 (,)cioo 13282 topGenctg 17376 Topctop 22813 sigAlgebracsiga 34091 sigaGencsigagen 34121 𝔅_ℝcbrsiga 34164 MblFnMcmbfm 34232 Probcprb 34391 rRndVarcrrv 34424 ∘_RV/𝑐corvc 34440

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 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-pre-lttri 11118 ax-pre-lttrn 11119

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 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-ioo 13286 df-topgen 17382 df-top 22814 df-bases 22866 df-esum 34011 df-siga 34092 df-sigagen 34122 df-brsiga 34165 df-meas 34179 df-mbfm 34233 df-prob 34392 df-rrv 34425 df-orvc 34441

This theorem is referenced by: orvcelval 34453 dstfrvel 34458 orvclteinc 34460

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