Step |
Hyp |
Ref |
Expression |
1 |
|
smfpreimale.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
2 |
|
smfpreimale.f |
⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
3 |
|
smfpreimale.d |
⊢ 𝐷 = dom 𝐹 |
4 |
|
smfpreimale.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
5 |
1 3
|
issmfle |
⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |
6 |
2 5
|
mpbid |
⊢ ( 𝜑 → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
7 |
6
|
simp3d |
⊢ ( 𝜑 → ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
8 |
|
breq2 |
⊢ ( 𝑎 = 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 ↔ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 ) ) |
9 |
8
|
rabbidv |
⊢ ( 𝑎 = 𝐴 → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } = { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ) |
10 |
9
|
eleq1d |
⊢ ( 𝑎 = 𝐴 → ( { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ↔ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
11 |
10
|
rspcva |
⊢ ( ( 𝐴 ∈ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
12 |
4 7 11
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) ) |