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 eqidd ⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) )
12 11 8 fvmpt2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
13 12 dmeqd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
14 nfcv ⊢ Ⅎ 𝑥 𝑍
15 14 nfcri ⊢ Ⅎ 𝑥 𝑛 ∈ 𝑍
16 2 15 nfan ⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑛 ∈ 𝑍 )
17 eqid ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
18 7 3expa ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
19 16 17 18 dmmptdf ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
20 13 19 eqtr2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
21 1 20 iineq2d ⊢ ( 𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
22 nfcv ⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 𝐴
23 nfmpt1 ⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
24 14 23 nfmpt ⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
25 nfcv ⊢ Ⅎ 𝑥 𝑛
26 24 25 nffv ⊢ Ⅎ 𝑥 ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
27 26 nfdm ⊢ Ⅎ 𝑥 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
28 14 27 nfiin ⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
29 22 28 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 } )
30 21 29 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 } )
31 nfv ⊢ Ⅎ 𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
32 3 31 nfan ⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
33 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
34 33 nfcri ⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
35 1 34 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
36 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
37 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
38 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
39 38 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
40 13 19 eqtrd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
41 40 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
42 39 41 eleqtrd ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
43 12 fveq1d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
44 43 3adant3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
45 simp3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
46 fvmpt4 ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
47 45 7 46 syl2anc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
48 44 47 eqtr2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
49 48 breq1d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 𝑦 ↔ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
50 36 37 42 49 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝐵 ≤ 𝑦 ↔ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 ) )
51 35 50 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 ) )
52 32 51 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 ) )
53 2 52 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 } )
54 30 53 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 } )
55 9 54 syl5eq ⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ≤ 𝑦 } )
56 nfcv ⊢ Ⅎ 𝑛 ℝ
57 nfra1 ⊢ Ⅎ 𝑛 ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
58 56 57 nfrex ⊢ Ⅎ 𝑛 ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦
59 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 𝐴
60 58 59 nfrabw ⊢ Ⅎ 𝑛 { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 }
61 9 60 nfcxfr ⊢ Ⅎ 𝑛 𝐷
62 61 nfcri ⊢ Ⅎ 𝑛 𝑥 ∈ 𝐷
63 1 62 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ 𝐷 )
64 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
65 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
66 rabidim1 ⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝐵 ≤ 𝑦 } → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
67 66 9 eleq2s ⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
68 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
69 67 68 sylan ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
70 69 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
71 64 65 70 48 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
72 63 71 mpteq2da ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
73 72 rneqd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
74 73 supeq1d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → sup ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
75 2 55 74 mpteq12da ⊢ ( 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 , < ) ) )
76 10 75 syl5eq ⊢ ( 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 , < ) ) )
77 nfmpt1 ⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
78 1 8 fmptd2f ⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
79 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 }
80 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 , < ) )
81 77 24 4 5 6 78 79 80 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 ) )
82 76 81 eqeltrd ⊢ ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) )