Metamath Proof Explorer


Theorem ovolval4lem2

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 , but here f is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses ovolval4lem2.a ( 𝜑𝐴 ⊆ ℝ )
ovolval4lem2.m 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) }
ovolval4lem2.g 𝐺 = ( 𝑛 ∈ ℕ ↦ ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ )
Assertion ovolval4lem2 ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) )

Proof

Step Hyp Ref Expression
1 ovolval4lem2.a ( 𝜑𝐴 ⊆ ℝ )
2 ovolval4lem2.m 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) }
3 ovolval4lem2.g 𝐺 = ( 𝑛 ∈ ℕ ↦ ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ )
4 iftrue ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) → if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) = ( 2nd ‘ ( 𝑓𝑛 ) ) )
5 4 opeq2d ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ = ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) ⟩ )
6 5 adantl ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ = ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) ⟩ )
7 df-br ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ↔ ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
8 7 biimpi ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
9 8 adantl ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
10 6 9 eqeltrd ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ≤ )
11 iffalse ( ¬ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) → if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) = ( 1st ‘ ( 𝑓𝑛 ) ) )
12 11 opeq2d ( ¬ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ = ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ⟩ )
13 12 adantl ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ = ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ⟩ )
14 elmapi ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ × ℝ ) )
15 14 ffvelrnda ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓𝑛 ) ∈ ( ℝ × ℝ ) )
16 xp1st ( ( 𝑓𝑛 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓𝑛 ) ) ∈ ℝ )
17 15 16 syl ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓𝑛 ) ) ∈ ℝ )
18 17 leidd ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 1st ‘ ( 𝑓𝑛 ) ) )
19 df-br ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 1st ‘ ( 𝑓𝑛 ) ) ↔ ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
20 18 19 sylib ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
21 20 adantr ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ⟩ ∈ ≤ )
22 13 21 eqeltrd ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ≤ )
23 10 22 pm2.61dan ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ≤ )
24 xp2nd ( ( 𝑓𝑛 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓𝑛 ) ) ∈ ℝ )
25 15 24 syl ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝑓𝑛 ) ) ∈ ℝ )
26 25 17 ifcld ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ∈ ℝ )
27 opelxpi ( ( ( 1st ‘ ( 𝑓𝑛 ) ) ∈ ℝ ∧ if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ∈ ℝ ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ( ℝ × ℝ ) )
28 17 26 27 syl2anc ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ( ℝ × ℝ ) )
29 23 28 elind ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ⟨ ( 1st ‘ ( 𝑓𝑛 ) ) , if ( ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 2nd ‘ ( 𝑓𝑛 ) ) , ( 1st ‘ ( 𝑓𝑛 ) ) ) ⟩ ∈ ( ≤ ∩ ( ℝ × ℝ ) ) )
30 29 3 fmptd ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) )
31 reex ℝ ∈ V
32 31 31 xpex ( ℝ × ℝ ) ∈ V
33 32 inex2 ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V
34 33 a1i ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V )
35 nnex ℕ ∈ V
36 35 a1i ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ℕ ∈ V )
37 34 36 elmapd ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ↔ 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) )
38 30 37 mpbird ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) )
39 38 adantr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) )
40 simpr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ran ( (,) ∘ 𝑓 ) ) → 𝐴 ran ( (,) ∘ 𝑓 ) )
41 rexpssxrxp ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* )
42 41 a1i ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) )
43 14 42 fssd ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ* × ℝ* ) )
44 2fveq3 ( 𝑘 = 𝑛 → ( 1st ‘ ( 𝑓𝑘 ) ) = ( 1st ‘ ( 𝑓𝑛 ) ) )
45 2fveq3 ( 𝑘 = 𝑛 → ( 2nd ‘ ( 𝑓𝑘 ) ) = ( 2nd ‘ ( 𝑓𝑛 ) ) )
46 44 45 breq12d ( 𝑘 = 𝑛 → ( ( 1st ‘ ( 𝑓𝑘 ) ) ≤ ( 2nd ‘ ( 𝑓𝑘 ) ) ↔ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) ) )
47 46 cbvrabv { 𝑘 ∈ ℕ ∣ ( 1st ‘ ( 𝑓𝑘 ) ) ≤ ( 2nd ‘ ( 𝑓𝑘 ) ) } = { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝑓𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓𝑛 ) ) }
48 43 3 47 ovolval4lem1 ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝐺 ) ∧ ( vol ∘ ( (,) ∘ 𝑓 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) )
49 48 simpld ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝐺 ) )
50 49 adantr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ran ( (,) ∘ 𝑓 ) ) → ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝐺 ) )
51 40 50 sseqtrd ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ran ( (,) ∘ 𝑓 ) ) → 𝐴 ran ( (,) ∘ 𝐺 ) )
52 51 adantrr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝐴 ran ( (,) ∘ 𝐺 ) )
53 simpr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) )
54 48 simprd ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( vol ∘ ( (,) ∘ 𝑓 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) )
55 coass ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( vol ∘ ( (,) ∘ 𝑓 ) )
56 55 a1i ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( vol ∘ ( (,) ∘ 𝑓 ) ) )
57 coass ( ( vol ∘ (,) ) ∘ 𝐺 ) = ( vol ∘ ( (,) ∘ 𝐺 ) )
58 57 a1i ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝐺 ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) )
59 54 56 58 3eqtr4d ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( ( vol ∘ (,) ) ∘ 𝐺 ) )
60 59 fveq2d ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) )
61 60 adantr ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) )
62 53 61 eqtrd ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) )
63 62 adantrl ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) )
64 52 63 jca ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → ( 𝐴 ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) )
65 coeq2 ( 𝑔 = 𝐺 → ( (,) ∘ 𝑔 ) = ( (,) ∘ 𝐺 ) )
66 65 rneqd ( 𝑔 = 𝐺 → ran ( (,) ∘ 𝑔 ) = ran ( (,) ∘ 𝐺 ) )
67 66 unieqd ( 𝑔 = 𝐺 ran ( (,) ∘ 𝑔 ) = ran ( (,) ∘ 𝐺 ) )
68 67 sseq2d ( 𝑔 = 𝐺 → ( 𝐴 ran ( (,) ∘ 𝑔 ) ↔ 𝐴 ran ( (,) ∘ 𝐺 ) ) )
69 coeq2 ( 𝑔 = 𝐺 → ( ( vol ∘ (,) ) ∘ 𝑔 ) = ( ( vol ∘ (,) ) ∘ 𝐺 ) )
70 69 fveq2d ( 𝑔 = 𝐺 → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) )
71 70 eqeq2d ( 𝑔 = 𝐺 → ( 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ↔ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) )
72 68 71 anbi12d ( 𝑔 = 𝐺 → ( ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ↔ ( 𝐴 ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) )
73 72 rspcev ( ( 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
74 39 64 73 syl2anc ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
75 74 rexlimiva ( ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
76 inss2 ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ )
77 mapss ( ( ( ℝ × ℝ ) ∈ V ∧ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) ) → ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) )
78 32 76 77 mp2an ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ )
79 78 sseli ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) )
80 79 adantr ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) )
81 simpr ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
82 coeq2 ( 𝑓 = 𝑔 → ( (,) ∘ 𝑓 ) = ( (,) ∘ 𝑔 ) )
83 82 rneqd ( 𝑓 = 𝑔 → ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝑔 ) )
84 83 unieqd ( 𝑓 = 𝑔 ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝑔 ) )
85 84 sseq2d ( 𝑓 = 𝑔 → ( 𝐴 ran ( (,) ∘ 𝑓 ) ↔ 𝐴 ran ( (,) ∘ 𝑔 ) ) )
86 coeq2 ( 𝑓 = 𝑔 → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( ( vol ∘ (,) ) ∘ 𝑔 ) )
87 86 fveq2d ( 𝑓 = 𝑔 → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) )
88 87 eqeq2d ( 𝑓 = 𝑔 → ( 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ↔ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
89 85 88 anbi12d ( 𝑓 = 𝑔 → ( ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) )
90 89 rspcev ( ( 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) )
91 80 81 90 syl2anc ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) )
92 91 rexlimiva ( ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) )
93 75 92 impbii ( ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) )
94 93 rabbii { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) } = { 𝑦 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) }
95 2 94 eqtri 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) }
96 1 95 ovolval3 ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) )