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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenuncl | Structured version Visualization version GIF version |
Description: The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenuncl.1 | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenuncl.2 | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenuncl.3 | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
caragenuncl.4 | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
Ref | Expression |
---|---|
caragenuncl | ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenuncl.1 | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | eqid 2651 | . 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
3 | caragenuncl.2 | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | caragenuncl.3 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
5 | 1, 3, 4, 2 | caragenelss 41036 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom 𝑂) |
6 | caragenuncl.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
7 | 1, 3, 6, 2 | caragenelss 41036 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom 𝑂) |
8 | 5, 7 | unssd 3822 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂) |
9 | 1, 2 | unidmex 39531 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
10 | ssexg 4837 | . . . . 5 ⊢ (((𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V) → (𝐸 ∪ 𝐹) ∈ V) | |
11 | 8, 9, 10 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V) |
12 | elpwg 4199 | . . . 4 ⊢ ((𝐸 ∪ 𝐹) ∈ V → ((𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂 ↔ (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂 ↔ (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂)) |
14 | 8, 13 | mpbird 247 | . 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂) |
15 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
16 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐸 ∈ 𝑆) |
17 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐹 ∈ 𝑆) |
18 | elpwi 4201 | . . . 4 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
19 | 18 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
20 | 15, 3, 16, 17, 2, 19 | caragenuncllem 41047 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝑎 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝑎)) |
21 | 1, 2, 3, 14, 20 | carageneld 41037 | 1 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∪ cun 3605 ⊆ wss 3607 𝒫 cpw 4191 ∪ cuni 4468 dom cdm 5143 ‘cfv 5926 OutMeascome 41024 CaraGenccaragen 41026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-addass 10039 ax-i2m1 10042 ax-1ne0 10043 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-xadd 11985 df-icc 12220 df-ome 41025 df-caragen 41027 |
This theorem is referenced by: caragenfiiuncl 41050 |
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