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| Mirrors > Home > MPE Home > Th. List > elmapd | Structured version Visualization version GIF version | ||
| Description: Deduction form of elmapg 8879. (Contributed by BJ, 11-Apr-2020.) |
| Ref | Expression |
|---|---|
| elmapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| elmapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| Ref | Expression |
|---|---|
| elmapd | ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elmapd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | elmapd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | elmapg 8879 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
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