| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fvres | Structured version Visualization version GIF version | ||
| Description: The value of a restricted function. (Contributed by NM, 2-Aug-1994.) |
| Ref | Expression |
|---|---|
| fvres | ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | brresi 6006 | . . . 4 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴𝐹𝑥)) |
| 3 | 2 | baib 535 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ 𝐴𝐹𝑥)) |
| 4 | 3 | iotabidv 6545 | . 2 ⊢ (𝐴 ∈ 𝐵 → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥)) |
| 5 | df-fv 6569 | . 2 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) | |
| 6 | df-fv 6569 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
| 7 | 4, 5, 6 | 3eqtr4g 2802 | 1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) |
| Copyright terms: Public domain | W3C validator |