Theorem caragenelss 45203

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 45197	. . . . 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 231	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥)))
6	5	simpld 495	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂)
7		caragenelss.x	. . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂
8	7	eqcomi 2741	. . . . 5 ⊢ ∪ dom 𝑂 = 𝑋
9	8	pweqi 4617	. . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
11	6, 10	eleqtrd 2835	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
12		elpwg 4604	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
13	1, 12	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
14	11, 13	mpbid 231	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  dom cdm 5675  ‘cfv 6540  (class class class)co 7405   +_𝑒 cxad 13086  OutMeascome 45191  CaraGenccaragen 45193

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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-caragen 45194

This theorem is referenced by:  caragenuncllem  45214  caragenuncl  45215