Step |
Hyp |
Ref |
Expression |
1 |
|
rnmptlb.1 |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ) |
2 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
3 |
2
|
elrnmpt |
⊢ ( 𝑧 ∈ V → ( 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
4 |
3
|
elv |
⊢ ( 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
5 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 |
6 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 ≤ 𝑧 |
7 |
|
rspa |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝑤 ≤ 𝐵 ) |
8 |
7
|
3adant3 |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑤 ≤ 𝐵 ) |
9 |
|
simp3 |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑧 = 𝐵 ) |
10 |
8 9
|
breqtrrd |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑤 ≤ 𝑧 ) |
11 |
10
|
3exp |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → ( 𝑥 ∈ 𝐴 → ( 𝑧 = 𝐵 → 𝑤 ≤ 𝑧 ) ) ) |
12 |
5 6 11
|
rexlimd |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ≤ 𝑧 ) ) |
13 |
12
|
imp |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → 𝑤 ≤ 𝑧 ) |
14 |
13
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → 𝑤 ≤ 𝑧 ) |
15 |
4 14
|
sylan2b |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) ∧ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → 𝑤 ≤ 𝑧 ) |
16 |
15
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) → ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑧 ) |
17 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵 ) ) |
18 |
17
|
ralbidv |
⊢ ( 𝑦 = 𝑤 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) ) |
19 |
18
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) |
20 |
1 19
|
sylib |
⊢ ( 𝜑 → ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ) |
21 |
16 20
|
reximddv3 |
⊢ ( 𝜑 → ∃ 𝑤 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑧 ) |
22 |
|
breq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧 ) ) |
23 |
22
|
ralbidv |
⊢ ( 𝑤 = 𝑦 → ( ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑧 ↔ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) ) |
24 |
23
|
cbvrexvw |
⊢ ( ∃ 𝑤 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑧 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) |
25 |
21 24
|
sylib |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) |