orvccel - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > orvccel		Structured version Visualization version GIF version

Theorem orvccel 34429

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 34427	. 2 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) = (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))
6	2	sgsiga 34108	. . 3 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
7		cldssbrsiga 34153	. . . . 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 4009	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (sigaGen‘𝐽))
11	1, 6, 3, 10	mbfmcnvima 34222	. 2 ⊢ (𝜑 → (^◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) ∈ 𝑆)
12	5, 11	eqeltrd 2844	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐𝑅𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 ∪ cuni 4931 class class class wbr 5166 ^◡ccnv 5699 ran crn 5701 “ cima 5703 ‘cfv 6575 (class class class)co 7450 Topctop 22922 Clsdccld 23047 sigAlgebracsiga 34074 sigaGencsigagen 34104 MblFnMcmbfm 34215 ∘_RV/𝑐corvc 34422

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-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-ac2 10534

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-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 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-pss 3996 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-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 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-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-oi 9581 df-dju 9972 df-card 10010 df-acn 10013 df-ac 10187 df-top 22923 df-cld 23050 df-siga 34075 df-sigagen 34105 df-mbfm 34216 df-orvc 34423

This theorem is referenced by: orrvccel 34433

W3C validator