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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 34305 | . 2 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) |
| 4 | ssrab2 4080 | . 2 ⊢ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ⊆ 𝒫 𝑂 | |
| 5 | 3, 4 | eqsstrdi 4028 | 1 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 +𝑒 cxad 13152 [,]cicc 13390 toCaraSigaccarsg 34303 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-carsg 34304 |
| This theorem is referenced by: carsguni 34310 elcarsgss 34311 carsggect 34320 carsgsiga 34324 omsmeas 34325 |
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