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 2819 | . 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 42774 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom 𝑂) |
6 | caragenuncl.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
7 | 1, 3, 6, 2 | caragenelss 42774 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom 𝑂) |
8 | 5, 7 | unssd 4160 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂) |
9 | 1, 2 | unidmex 41303 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
10 | ssexg 5218 | . . . . 5 ⊢ (((𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V) → (𝐸 ∪ 𝐹) ∈ V) | |
11 | 8, 9, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V) |
12 | elpwg 4543 | . . . 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 259 | . 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝒫 ∪ dom 𝑂) |
15 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
16 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐸 ∈ 𝑆) |
17 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝐹 ∈ 𝑆) |
18 | elpwi 4549 | . . . 4 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
19 | 18 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
20 | 15, 3, 16, 17, 2, 19 | caragenuncllem 42785 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝑎 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝑎)) |
21 | 1, 2, 3, 14, 20 | carageneld 42775 | 1 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ∪ cun 3932 ⊆ wss 3934 𝒫 cpw 4537 ∪ cuni 4830 dom cdm 5548 ‘cfv 6348 OutMeascome 42762 CaraGenccaragen 42764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-addass 10594 ax-i2m1 10597 ax-rnegex 10600 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-xadd 12500 df-icc 12737 df-ome 42763 df-caragen 42765 |
This theorem is referenced by: caragenfiiuncl 42788 |
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