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| Mirrors > Home > MPE Home > Th. List > ffvelcdmd | Structured version Visualization version GIF version | ||
| Description: A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
| Ref | Expression |
|---|---|
| ffvelcdmd.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| ffvelcdmd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| ffvelcdmd | ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ffvelcdmd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
| 2 | ffvelcdmd.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 3 | 2 | ffvelcdmda 7104 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
| 4 | 1, 3 | mpdan 687 | 1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐵) |
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