Theorem caragenelss 46917

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 46911	. . . . 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 2744	. . . . 5 ⊢ ∪ dom 𝑂 = 𝑋
9	8	pweqi 4547	. . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
11	6, 10	eleqtrd 2837	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
12		elpwg 4534	. . 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 1542   ∈ wcel 2114  ∀wral 3049   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  𝒫 cpw 4531  ∪ cuni 4840  dom cdm 5620  ‘cfv 6487  (class class class)co 7356   +_𝑒 cxad 13050  OutMeascome 46905  CaraGenccaragen 46907

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6443  df-fun 6489  df-fv 6495  df-ov 7359  df-caragen 46908

This theorem is referenced by:  caragenuncllem  46928  caragenuncl  46929