orvccel - Mathbox for Thierry Arnoux

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Theorem orvccel 31715

Description: If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)

**Hypotheses**
Ref	Expression
orvccel.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
orvccel.2	⊢ (𝜑 → 𝐽 ∈ Top)
orvccel.3	⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
orvccel.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
orvccel.5	⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (Clsd‘𝐽))

**Assertion**
Ref	Expression
orvccel	⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) ∈ 𝑆)

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

**Proof of Theorem orvccel**
Step	Hyp	Ref	Expression
1		orvccel.1	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2		orvccel.2	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
3		orvccel.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
4		orvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	1, 2, 3, 4	orvcval4 31713	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))
6	2	sgsiga 31396	. . 3 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
7		cldssbrsiga 31441	. . . . 5 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
8	2, 7	syl 17	. . . 4 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
9		orvccel.5	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (Clsd‘𝐽))
10	8, 9	sseldd 3967	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (sigaGen‘𝐽))
11	1, 6, 3, 10	mbfmcnvima 31510	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) ∈ 𝑆)
12	5, 11	eqeltrd 2913	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ∪ cuni 4831 class class class wbr 5058 ^◡ccnv 5548 ran crn 5550 “ cima 5552 ‘cfv 6349 (class class class)co 7150 Topctop 21495 Clsdccld 21618 sigAlgebracsiga 31362 sigaGencsigagen 31392 MblFnMcmbfm 31503 ∘_RV/𝑐corvc 31708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-ac2 9879

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-ac 9536 df-top 21496 df-cld 21621 df-siga 31363 df-sigagen 31393 df-mbfm 31504 df-orvc 31709

This theorem is referenced by: orrvccel 31719

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