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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcoel | Structured version Visualization version GIF version |
Description: If the relation produces open 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) |
orvcoel.5 | ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ 𝐽) |
Ref | Expression |
---|---|
orvcoel | ⊢ (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 34211 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴})) |
6 | 2 | sgsiga 33892 | . . 3 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
7 | sssigagen 33895 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (sigaGen‘𝐽)) | |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ (sigaGen‘𝐽)) |
9 | orvcoel.5 | . . . 4 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ 𝐽) | |
10 | 8, 9 | sseldd 3977 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (sigaGen‘𝐽)) |
11 | 1, 6, 3, 10 | mbfmcnvima 34006 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2825 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 {crab 3418 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 ◡ccnv 5677 ran crn 5679 “ cima 5681 ‘cfv 6549 (class class class)co 7419 Topctop 22839 sigAlgebracsiga 33858 sigaGencsigagen 33888 MblFnMcmbfm 33999 ∘RV/𝑐corvc 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 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-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fo 6555 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-siga 33859 df-sigagen 33889 df-mbfm 34000 df-orvc 34207 |
This theorem is referenced by: orrvcoel 34216 |
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