Theorem caragenelss 46502

Description: An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caragenelss.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragenelss.s	⊢ 𝑆 = (CaraGen‘𝑂)
caragenelss.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caragenelss.x	⊢ 𝑋 = ∪ dom 𝑂

Assertion

Ref	Expression
caragenelss	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Proof of Theorem caragenelss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caragenelss.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
2		caragenelss.o	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		caragenelss.s	. . . . . 6 ⊢ 𝑆 = (CaraGen‘𝑂)
4	2, 3	caragenel 46496	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥))))
5	1, 4	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥)))
6	5	simpld 494	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂)
7		caragenelss.x	. . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂
8	7	eqcomi 2738	. . . . 5 ⊢ ∪ dom 𝑂 = 𝑋
9	8	pweqi 4567	. . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
11	6, 10	eleqtrd 2830	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
12		elpwg 4554	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
13	1, 12	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
14	11, 13	mpbid 232	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  dom cdm 5619  ‘cfv 6482  (class class class)co 7349   +_𝑒 cxad 13012  OutMeascome 46490  CaraGenccaragen 46492

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-caragen 46493

This theorem is referenced by:  caragenuncllem  46513  caragenuncl  46514