| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > df-case | GIF version | ||
| Description: The "case" construction: if 𝐹:𝐴⟶𝑋 and 𝐺:𝐵⟶𝑋 are functions, then case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋 is the natural function obtained by a definition by cases, hence the name. It is the unique function whose existence is asserted by the universal property of disjoint unions updjud 7281. The definition is adapted to make sense also for binary relations (where the universal property also holds). (Contributed by MC and BJ, 10-Jul-2022.) |
| Ref | Expression |
|---|---|
| df-case | ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cR | . . 3 class 𝑅 | |
| 2 | cS | . . 3 class 𝑆 | |
| 3 | 1, 2 | cdjucase 7282 | . 2 class case(𝑅, 𝑆) |
| 4 | cinl 7244 | . . . . 5 class inl | |
| 5 | 4 | ccnv 4724 | . . . 4 class ◡inl |
| 6 | 1, 5 | ccom 4729 | . . 3 class (𝑅 ∘ ◡inl) |
| 7 | cinr 7245 | . . . . 5 class inr | |
| 8 | 7 | ccnv 4724 | . . . 4 class ◡inr |
| 9 | 2, 8 | ccom 4729 | . . 3 class (𝑆 ∘ ◡inr) |
| 10 | 6, 9 | cun 3198 | . 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
| 11 | 3, 10 | wceq 1397 | 1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
| Colors of variables: wff set class |
| This definition is referenced by: casefun 7284 casedm 7285 caserel 7286 caseinj 7288 caseinl 7290 caseinr 7291 |
| Copyright terms: Public domain | W3C validator |