| 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 2737 | . 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 46516 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom 𝑂) |
| 6 | caragenuncl.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
| 7 | 1, 3, 6, 2 | caragenelss 46516 | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom 𝑂) |
| 8 | 5, 7 | unssd 4192 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂) |
| 9 | 1, 2 | unidmex 45055 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
| 10 | ssexg 5323 | . . . . 5 ⊢ (((𝐸 ∪ 𝐹) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V) → (𝐸 ∪ 𝐹) ∈ V) | |
| 11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V) |
| 12 | elpwg 4603 | . . . 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 257 | . 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 4607 | . . . 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 46527 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝑎 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝑎)) |
| 21 | 1, 2, 3, 14, 20 | carageneld 46517 | 1 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 OutMeascome 46504 CaraGenccaragen 46506 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-addass 11220 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 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-po 5592 df-so 5593 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-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-xadd 13155 df-icc 13394 df-ome 46505 df-caragen 46507 |
| This theorem is referenced by: caragenfiiuncl 46530 |
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