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| Mirrors > Home > MPE Home > Th. List > df-mpt | Structured version Visualization version GIF version | ||
| Description: Define maps-to notation for defining a function via a rule. Read as "the function which maps 𝑥 (in 𝐴) to 𝐵(𝑥)". The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. An example is the square function for complex numbers, (𝑥 ∈ ℂ ↦ (𝑥↑2)). Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.) |
| Ref | Expression |
|---|---|
| df-mpt | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vx | . . 3 setvar 𝑥 | |
| 2 | cA | . . 3 class 𝐴 | |
| 3 | cB | . . 3 class 𝐵 | |
| 4 | 1, 2, 3 | cmpt 5225 | . 2 class (𝑥 ∈ 𝐴 ↦ 𝐵) |
| 5 | 1 | cv 1539 | . . . . 5 class 𝑥 |
| 6 | 5, 2 | wcel 2108 | . . . 4 wff 𝑥 ∈ 𝐴 |
| 7 | vy | . . . . . 6 setvar 𝑦 | |
| 8 | 7 | cv 1539 | . . . . 5 class 𝑦 |
| 9 | 8, 3 | wceq 1540 | . . . 4 wff 𝑦 = 𝐵 |
| 10 | 6, 9 | wa 395 | . . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) |
| 11 | 10, 1, 7 | copab 5205 | . 2 class {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
| 12 | 4, 11 | wceq 1540 | 1 wff (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
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