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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgcl | Structured version Visualization version GIF version |
Description: Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
Ref | Expression |
---|---|
carsgcl | ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
2 | carsgval.2 | . . 3 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
3 | 1, 2 | carsgval 34268 | . 2 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) |
4 | ssrab2 4103 | . 2 ⊢ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ⊆ 𝒫 𝑂 | |
5 | 3, 4 | eqsstrdi 4063 | 1 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 0cc0 11184 +∞cpnf 11321 +𝑒 cxad 13173 [,]cicc 13410 toCaraSigaccarsg 34266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-carsg 34267 |
This theorem is referenced by: carsguni 34273 elcarsgss 34274 carsggect 34283 carsgsiga 34287 omsmeas 34288 |
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