Metamath Proof Explorer

Theorem smfsupmpt

Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of Fremlin1 p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses smfsupmpt.n ⊢ Ⅎ 𝑛 𝜑
smfsupmpt.x ⊢ Ⅎ 𝑥 𝜑
smfsupmpt.y ⊢ Ⅎ 𝑦 𝜑
smfsupmpt.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
smfsupmpt.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
smfsupmpt.s ⊢ ( 𝜑 → 𝑆 ∈ SAlg )
smfsupmpt.b ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
smfsupmpt.f ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
smfsupmpt.d ⊢ 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 }
smfsupmpt.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐷 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) )
Assertion smfsupmpt ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 smfsupmpt.n ⊢ Ⅎ 𝑛 𝜑
2 smfsupmpt.x ⊢ Ⅎ 𝑥 𝜑
3 smfsupmpt.y ⊢ Ⅎ 𝑦 𝜑
4 smfsupmpt.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
5 smfsupmpt.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
6 smfsupmpt.s ⊢ ( 𝜑 → 𝑆 ∈ SAlg )
7 smfsupmpt.b ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
8 smfsupmpt.f ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
9 smfsupmpt.d ⊢ 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 }
10 smfsupmpt.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐷 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) )
11 10 a1i ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐷 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) )
12 9 a1i ⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } )
13 eqidd ⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) )
14 13 8 fvmpt2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
15 14 dmeqd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
16 nfcv ⊢ Ⅎ 𝑥 𝑛
17 nfcv ⊢ Ⅎ 𝑥 𝑍
18 16 17 nfel ⊢ Ⅎ 𝑥 𝑛 ∈ 𝑍
19 2 18 nfan ⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑛 ∈ 𝑍 )
20 eqid ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
21 6 adantr ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑆 ∈ SAlg )
22 7 3expa ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
23 19 21 22 8 smffmpt ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℝ )
24 23 fvmptelrn ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
25 19 20 24 dmmptdf ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
26 eqidd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = 𝐴 )
27 15 25 26 3eqtrrd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
28 1 27 iineq2d ⊢ ( 𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
29 nfcv ⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 𝐴
30 nfmpt1 ⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
31 17 30 nfmpt ⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
32 31 16 nffv ⊢ Ⅎ 𝑥 ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
33 32 nfdm ⊢ Ⅎ 𝑥 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
34 17 33 nfiin ⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
35 29 34 rabeqf ⊢ ( ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } )
36 28 35 syl ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } )
37 nfv ⊢ Ⅎ 𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
38 3 37 nfan ⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
39 nfcv ⊢ Ⅎ 𝑛 𝑥
40 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
41 39 40 nfel ⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
42 1 41 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
43 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
44 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
45 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
46 45 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
47 27 eqcomd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
48 47 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
49 46 48 eleqtrd ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
50 14 fveq1d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
51 50 3adant3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
52 simp3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
53 20 fvmpt2 ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
54 52 7 53 syl2anc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
55 51 54 eqtr2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
56 55 breq1d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 𝑦 ↔ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
57 43 44 49 56 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝐵 ≤ 𝑦 ↔ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
58 42 57 ralbida ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
59 38 58 rexbid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
60 2 59 rabbida ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } )
61 36 60 eqtrd ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } )
62 12 61 eqtrd ⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } )
63 2 62 alrimi ⊢ ( 𝜑 → ∀ 𝑥 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } )
64 nfcv ⊢ Ⅎ 𝑛 ℝ
65 nfra1 ⊢ Ⅎ 𝑛 ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
66 64 65 nfrex ⊢ Ⅎ 𝑛 ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
67 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 𝐴
68 66 67 nfrabw ⊢ Ⅎ 𝑛 { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 }
69 9 68 nfcxfr ⊢ Ⅎ 𝑛 𝐷
70 39 69 nfel ⊢ Ⅎ 𝑛 𝑥 ∈ 𝐷
71 1 70 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ 𝐷 )
72 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
73 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
74 9 eleq2i ⊢ ( 𝑥 ∈ 𝐷 ↔ 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } )
75 74 biimpi ⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } )
76 rabidim1 ⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
77 75 76 syl ⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
78 77 adantr ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
79 simpr ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
80 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
81 78 79 80 syl2anc ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
82 81 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
83 55 idi ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
84 72 73 82 83 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
85 71 84 mpteq2da ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
86 85 rneqd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
87 86 supeq1d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
88 87 ex ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 → sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
89 2 88 ralrimi ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
90 mpteq12f ⊢ ( ( ∀ 𝑥 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ∧ ∀ 𝑥 ∈ 𝐷 sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) → ( 𝑥 ∈ 𝐷 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
91 63 89 90 syl2anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
92 11 91 eqtrd ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
93 nfmpt1 ⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
94 eqid ⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
95 1 8 94 fmptdf ⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
96 eqid ⊢ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 }
97 eqid ⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
98 93 31 4 5 6 95 96 97 smfsup ⊢ ( 𝜑 → ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ∈ ( SMblFn ‘ 𝑆 ) )
99 92 98 eqeltrd ⊢ ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) )