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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 | β’ (π β π β βͺ ran sigAlgebra) |
orvccel.2 | β’ (π β π½ β Top) |
orvccel.3 | β’ (π β π β (πMblFnM(sigaGenβπ½))) |
orvccel.4 | β’ (π β π΄ β π) |
orvcoel.5 | β’ (π β {π¦ β βͺ π½ β£ π¦π π΄} β π½) |
Ref | Expression |
---|---|
orvcoel | β’ (π β (πβRV/ππ π΄) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvccel.1 | . . 3 β’ (π β π β βͺ ran sigAlgebra) | |
2 | orvccel.2 | . . 3 β’ (π β π½ β Top) | |
3 | orvccel.3 | . . 3 β’ (π β π β (πMblFnM(sigaGenβπ½))) | |
4 | orvccel.4 | . . 3 β’ (π β π΄ β π) | |
5 | 1, 2, 3, 4 | orvcval4 33459 | . 2 β’ (π β (πβRV/ππ π΄) = (β‘π β {π¦ β βͺ π½ β£ π¦π π΄})) |
6 | 2 | sgsiga 33140 | . . 3 β’ (π β (sigaGenβπ½) β βͺ ran sigAlgebra) |
7 | sssigagen 33143 | . . . . 5 β’ (π½ β Top β π½ β (sigaGenβπ½)) | |
8 | 2, 7 | syl 17 | . . . 4 β’ (π β π½ β (sigaGenβπ½)) |
9 | orvcoel.5 | . . . 4 β’ (π β {π¦ β βͺ π½ β£ π¦π π΄} β π½) | |
10 | 8, 9 | sseldd 3984 | . . 3 β’ (π β {π¦ β βͺ π½ β£ π¦π π΄} β (sigaGenβπ½)) |
11 | 1, 6, 3, 10 | mbfmcnvima 33254 | . 2 β’ (π β (β‘π β {π¦ β βͺ π½ β£ π¦π π΄}) β π) |
12 | 5, 11 | eqeltrd 2834 | 1 β’ (π β (πβRV/ππ π΄) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 {crab 3433 β wss 3949 βͺ cuni 4909 class class class wbr 5149 β‘ccnv 5676 ran crn 5678 β cima 5680 βcfv 6544 (class class class)co 7409 Topctop 22395 sigAlgebracsiga 33106 sigaGencsigagen 33136 MblFnMcmbfm 33247 βRV/πcorvc 33454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-siga 33107 df-sigagen 33137 df-mbfm 33248 df-orvc 33455 |
This theorem is referenced by: orrvcoel 33464 |
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