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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenuni | Structured version Visualization version GIF version |
Description: The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenuni.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenuni.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragenuni | ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenuni.o | . . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | caragenuni.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
3 | 2 | caragenss 43671 | . . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂) |
5 | 4 | unissd 4819 | . 2 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂) |
6 | eqid 2734 | . . . 4 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
7 | 1, 6, 2 | caragenunidm 43675 | . . 3 ⊢ (𝜑 → ∪ dom 𝑂 ∈ 𝑆) |
8 | elssuni 4841 | . . 3 ⊢ (∪ dom 𝑂 ∈ 𝑆 → ∪ dom 𝑂 ⊆ ∪ 𝑆) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → ∪ dom 𝑂 ⊆ ∪ 𝑆) |
10 | 5, 9 | eqssd 3908 | 1 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 ∪ cuni 4809 dom cdm 5540 ‘cfv 6369 OutMeascome 43656 CaraGenccaragen 43658 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-xadd 12688 df-icc 12925 df-ome 43657 df-caragen 43659 |
This theorem is referenced by: caragendifcl 43681 carageniuncl 43690 unidmvon 43784 |
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