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| Mirrors > Home > MPE Home > Th. List > elmap | Structured version Visualization version GIF version | ||
| Description: Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.) |
| Ref | Expression |
|---|---|
| elmap.1 | ⊢ 𝐴 ∈ V |
| elmap.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| elmap | ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elmap.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | elmap.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | elmapg 8879 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴) |
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