Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvccel | Structured version Visualization version GIF version |
Description: If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
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)) |
Ref | Expression |
---|---|
orvccel | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ 𝑆) |
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 32093 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) |
6 | 2 | sgsiga 31776 | . . 3 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
7 | cldssbrsiga 31821 | . . . . 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 3888 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (sigaGen‘𝐽)) |
11 | 1, 6, 3, 10 | mbfmcnvima 31890 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2831 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 {crab 3055 ⊆ wss 3853 ∪ cuni 4805 class class class wbr 5039 ◡ccnv 5535 ran crn 5537 “ cima 5539 ‘cfv 6358 (class class class)co 7191 Topctop 21744 Clsdccld 21867 sigAlgebracsiga 31742 sigaGencsigagen 31772 MblFnMcmbfm 31883 ∘RV/𝑐corvc 32088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-ac2 10042 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-oi 9104 df-dju 9482 df-card 9520 df-acn 9523 df-ac 9695 df-top 21745 df-cld 21870 df-siga 31743 df-sigagen 31773 df-mbfm 31884 df-orvc 32089 |
This theorem is referenced by: orrvccel 32099 |
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