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Mirrors > Home > MPE Home > Th. List > ffvelrn | Structured version Visualization version GIF version |
Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
Ref | Expression |
---|---|
ffvelrn | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6545 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfvelrn 6901 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 583 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) |
4 | frn 6552 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 3900 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
6 | 5 | adantr 484 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
7 | 3, 6 | mpd 15 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
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