Metamath Proof Explorer


Theorem omessle

Description: The outer measure of a set is greater than or equal to the measure of a subset, Definition 113A (ii) of Fremlin1 p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses omessle.o ( 𝜑𝑂 ∈ OutMeas )
omessle.x 𝑋 = dom 𝑂
omessle.b ( 𝜑𝐵𝑋 )
omessle.a ( 𝜑𝐴𝐵 )
Assertion omessle ( 𝜑 → ( 𝑂𝐴 ) ≤ ( 𝑂𝐵 ) )

Proof

Step Hyp Ref Expression
1 omessle.o ( 𝜑𝑂 ∈ OutMeas )
2 omessle.x 𝑋 = dom 𝑂
3 omessle.b ( 𝜑𝐵𝑋 )
4 omessle.a ( 𝜑𝐴𝐵 )
5 1 2 unidmex ( 𝜑𝑋 ∈ V )
6 5 3 ssexd ( 𝜑𝐵 ∈ V )
7 6 4 ssexd ( 𝜑𝐴 ∈ V )
8 elpwg ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵𝐴𝐵 ) )
9 7 8 syl ( 𝜑 → ( 𝐴 ∈ 𝒫 𝐵𝐴𝐵 ) )
10 4 9 mpbird ( 𝜑𝐴 ∈ 𝒫 𝐵 )
11 3 2 sseqtrdi ( 𝜑𝐵 dom 𝑂 )
12 elpwg ( 𝐵 ∈ V → ( 𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂 ) )
13 6 12 syl ( 𝜑 → ( 𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂 ) )
14 11 13 mpbird ( 𝜑𝐵 ∈ 𝒫 dom 𝑂 )
15 isome ( 𝑂 ∈ OutMeas → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂𝑦 ) ) ) ) ) )
16 1 15 syl ( 𝜑 → ( 𝑂 ∈ OutMeas ↔ ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂𝑦 ) ) ) ) ) )
17 1 16 mpbid ( 𝜑 → ( ( ( ( 𝑂 : dom 𝑂 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑂 = 𝒫 dom 𝑂 ) ∧ ( 𝑂 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂 ( 𝑦 ≼ ω → ( 𝑂 𝑦 ) ≤ ( Σ^ ‘ ( 𝑂𝑦 ) ) ) ) )
18 17 simplrd ( 𝜑 → ∀ 𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) )
19 pweq ( 𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵 )
20 19 raleqdv ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ) )
21 fveq2 ( 𝑦 = 𝐵 → ( 𝑂𝑦 ) = ( 𝑂𝐵 ) )
22 21 breq2d ( 𝑦 = 𝐵 → ( ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ↔ ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) ) )
23 22 ralbidv ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) ) )
24 20 23 bitrd ( 𝑦 = 𝐵 → ( ∀ 𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ↔ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) ) )
25 24 rspcva ( ( 𝐵 ∈ 𝒫 dom 𝑂 ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦 ( 𝑂𝑧 ) ≤ ( 𝑂𝑦 ) ) → ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) )
26 14 18 25 syl2anc ( 𝜑 → ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) )
27 fveq2 ( 𝑧 = 𝐴 → ( 𝑂𝑧 ) = ( 𝑂𝐴 ) )
28 27 breq1d ( 𝑧 = 𝐴 → ( ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) ↔ ( 𝑂𝐴 ) ≤ ( 𝑂𝐵 ) ) )
29 28 rspcva ( ( 𝐴 ∈ 𝒫 𝐵 ∧ ∀ 𝑧 ∈ 𝒫 𝐵 ( 𝑂𝑧 ) ≤ ( 𝑂𝐵 ) ) → ( 𝑂𝐴 ) ≤ ( 𝑂𝐵 ) )
30 10 26 29 syl2anc ( 𝜑 → ( 𝑂𝐴 ) ≤ ( 𝑂𝐵 ) )