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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 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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