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| Mirrors > Home > MPE Home > Th. List > elmapi | Structured version Visualization version GIF version | ||
| Description: A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
| Ref | Expression |
|---|---|
| elmapi | ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elmapex 8888 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
| 2 | elmapg 8879 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ (𝐵 ↑m 𝐶) ↔ 𝐴:𝐶⟶𝐵)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐴 ∈ (𝐵 ↑m 𝐶) ↔ 𝐴:𝐶⟶𝐵)) |
| 4 | 3 | ibi 267 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) |
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