Metamath Proof Explorer

Theorem omscl

Description: A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019)

Ref Expression
Assertion omscl ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ⊆ ( 0 [,] +∞ ) )

Proof

Step Hyp Ref Expression
1 vex ⊢ 𝑥 ∈ V
2 simp2 ⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) )
3 2 ad2antrr ⊢ ( ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑦 ∈ 𝑥 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) )
4 ssrab2 ⊢ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ⊆ 𝒫 dom 𝑅
5 simpr ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } )
6 4 5 sseldi ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → 𝑥 ∈ 𝒫 dom 𝑅 )
7 fdm ⊢ ( 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) → dom 𝑅 = 𝑄 )
8 7 pweqd ⊢ ( 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) → 𝒫 dom 𝑅 = 𝒫 𝑄 )
9 2 8 syl ⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → 𝒫 dom 𝑅 = 𝒫 𝑄 )
10 9 adantr ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → 𝒫 dom 𝑅 = 𝒫 𝑄 )
11 6 10 eleqtrd ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → 𝑥 ∈ 𝒫 𝑄 )
12 elpwi ⊢ ( 𝑥 ∈ 𝒫 𝑄 → 𝑥 ⊆ 𝑄 )
13 11 12 syl ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → 𝑥 ⊆ 𝑄 )
14 13 sselda ⊢ ( ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑄 )
15 3 14 ffvelrnd ⊢ ( ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) )
16 15 ralrimiva ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → ∀ 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) )
17 nfcv ⊢ Ⅎ 𝑦 𝑥
18 17 esumcl ⊢ ( ( 𝑥 ∈ V ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) )
19 1 16 18 sylancr ⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) → Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) )
20 19 ralrimiva ⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → ∀ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) )
21 eqid ⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) )
22 21 rnmptss ⊢ ( ∀ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ⊆ ( 0 [,] +∞ ) )
23 20 22 syl ⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ⊆ ( 0 [,] +∞ ) )