Metamath Proof Explorer

Theorem smfinfmpt

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

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

Proof

Step Hyp Ref Expression
1 smfinfmpt.n ⊢ Ⅎ 𝑛 𝜑
2 smfinfmpt.x ⊢ Ⅎ 𝑥 𝜑
3 smfinfmpt.y ⊢ Ⅎ 𝑦 𝜑
4 smfinfmpt.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
5 smfinfmpt.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
6 smfinfmpt.s ⊢ ( 𝜑 → 𝑆 ∈ SAlg )
7 smfinfmpt.b ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
8 smfinfmpt.f ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
9 smfinfmpt.d ⊢ 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 }
10 smfinfmpt.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐷 ↦ inf ( 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 2 21 rabeqd ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } )
23 nfv ⊢ Ⅎ 𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
24 3 23 nfan ⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
25 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
26 25 nfcri ⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 )
27 1 26 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
28 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
29 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
30 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
31 30 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) )
32 13 19 eqtrd ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
33 32 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
34 31 33 eleqtrd ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
35 12 fveq1d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
36 35 3adant3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) )
37 simp3 ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
38 fvmpt4 ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
39 37 7 38 syl2anc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 )
40 36 39 eqtr2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
41 40 breq2d ⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
42 28 29 34 41 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
43 27 42 ralbida ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
44 24 43 rexbid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
45 2 44 rabbida ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
46 22 45 eqtrd ⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
47 9 46 eqtrid ⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
48 nfcv ⊢ Ⅎ 𝑛 ℝ
49 nfra1 ⊢ Ⅎ 𝑛 ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵
50 48 49 nfrexw ⊢ Ⅎ 𝑛 ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵
51 nfii1 ⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 𝐴
52 50 51 nfrabw ⊢ Ⅎ 𝑛 { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 }
53 9 52 nfcxfr ⊢ Ⅎ 𝑛 𝐷
54 53 nfcri ⊢ Ⅎ 𝑛 𝑥 ∈ 𝐷
55 1 54 nfan ⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ 𝐷 )
56 simpll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 )
57 simpr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
58 rabidim1 ⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
59 58 9 eleq2s ⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 )
60 eliinid ⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
61 59 60 sylan ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
62 61 adantll ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 )
63 56 57 62 40 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
64 55 63 mpteq2da ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
65 64 rneqd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
66 65 infeq1d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
67 2 47 66 mpteq12da ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
68 10 67 eqtrid ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
69 nfmpt1 ⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
70 nfmpt1 ⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
71 14 70 nfmpt ⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
72 1 8 fmptd2f ⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
73 eqid ⊢ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) }
74 eqid ⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( 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 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
75 69 71 4 5 6 72 73 74 smfinf ⊢ ( 𝜑 → ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ∈ ( SMblFn ‘ 𝑆 ) )
76 68 75 eqeltrd ⊢ ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) )