![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 32894 | . 2 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) |
4 | ssrab2 4037 | . 2 ⊢ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ⊆ 𝒫 𝑂 | |
5 | 3, 4 | eqsstrdi 3998 | 1 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 0cc0 11050 +∞cpnf 11185 +𝑒 cxad 13030 [,]cicc 13266 toCaraSigaccarsg 32892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-carsg 32893 |
This theorem is referenced by: carsguni 32899 elcarsgss 32900 carsggect 32909 carsgsiga 32913 omsmeas 32914 |
Copyright terms: Public domain | W3C validator |