orvccel - Mathbox for Thierry Arnoux

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Theorem orvccel 34082

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 34080	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))
6	2	sgsiga 33761	. . 3 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
7		cldssbrsiga 33806	. . . . 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 3981	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (sigaGen‘𝐽))
11	1, 6, 3, 10	mbfmcnvima 33875	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) ∈ 𝑆)
12	5, 11	eqeltrd 2829	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2099 {crab 3429 ⊆ wss 3947 ∪ cuni 4908 class class class wbr 5148 ^◡ccnv 5677 ran crn 5679 “ cima 5681 ‘cfv 6548 (class class class)co 7420 Topctop 22808 Clsdccld 22933 sigAlgebracsiga 33727 sigaGencsigagen 33757 MblFnMcmbfm 33868 ∘_RV/𝑐corvc 34075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-ac2 10487

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9534 df-dju 9925 df-card 9963 df-acn 9966 df-ac 10140 df-top 22809 df-cld 22936 df-siga 33728 df-sigagen 33758 df-mbfm 33869 df-orvc 34076

This theorem is referenced by: orrvccel 34086

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