Step |
Hyp |
Ref |
Expression |
1 |
|
supxrrernmpt.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
supxrrernmpt.a |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
3 |
|
supxrrernmpt.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
4 |
|
supxrrernmpt.y |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) |
5 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
6 |
1 5 3
|
rnmptssd |
⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
7 |
1 3 5 2
|
rnmptn0 |
⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ ) |
8 |
1 4
|
rnmptbdd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑧 ≤ 𝑦 ) |
9 |
|
supxrre |
⊢ ( ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ∧ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑧 ≤ 𝑦 ) → sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) = sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) ) |
10 |
6 7 8 9
|
syl3anc |
⊢ ( 𝜑 → sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) = sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) ) |