Step |
Hyp |
Ref |
Expression |
1 |
|
pimltmnf2.1 |
⊢ Ⅎ 𝑥 𝐹 |
2 |
|
pimltmnf2.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ ) |
3 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
4 |
|
nfcv |
⊢ Ⅎ 𝑦 𝐴 |
5 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 ) < -∞ |
6 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
7 |
1 6
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑥 < |
9 |
|
nfcv |
⊢ Ⅎ 𝑥 -∞ |
10 |
7 8 9
|
nfbr |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) < -∞ |
11 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
12 |
11
|
breq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) < -∞ ↔ ( 𝐹 ‘ 𝑦 ) < -∞ ) ) |
13 |
3 4 5 10 12
|
cbvrabw |
⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < -∞ } = { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) < -∞ } |
14 |
13
|
a1i |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < -∞ } = { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) < -∞ } ) |
15 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
16 |
15
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → -∞ ∈ ℝ* ) |
17 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
18 |
17
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ* ) |
19 |
17
|
mnfltd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → -∞ < ( 𝐹 ‘ 𝑦 ) ) |
20 |
16 18 19
|
xrltled |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → -∞ ≤ ( 𝐹 ‘ 𝑦 ) ) |
21 |
16 18
|
xrlenltd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( -∞ ≤ ( 𝐹 ‘ 𝑦 ) ↔ ¬ ( 𝐹 ‘ 𝑦 ) < -∞ ) ) |
22 |
20 21
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ¬ ( 𝐹 ‘ 𝑦 ) < -∞ ) |
23 |
22
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ¬ ( 𝐹 ‘ 𝑦 ) < -∞ ) |
24 |
|
rabeq0 |
⊢ ( { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) < -∞ } = ∅ ↔ ∀ 𝑦 ∈ 𝐴 ¬ ( 𝐹 ‘ 𝑦 ) < -∞ ) |
25 |
23 24
|
sylibr |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) < -∞ } = ∅ ) |
26 |
14 25
|
eqtrd |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < -∞ } = ∅ ) |