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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 6935. 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 6936 | . 2 class case(𝑅, 𝑆) |
4 | cinl 6898 | . . . . 5 class inl | |
5 | 4 | ccnv 4508 | . . . 4 class ◡inl |
6 | 1, 5 | ccom 4513 | . . 3 class (𝑅 ∘ ◡inl) |
7 | cinr 6899 | . . . . 5 class inr | |
8 | 7 | ccnv 4508 | . . . 4 class ◡inr |
9 | 2, 8 | ccom 4513 | . . 3 class (𝑆 ∘ ◡inr) |
10 | 6, 9 | cun 3039 | . 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
11 | 3, 10 | wceq 1316 | 1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
Colors of variables: wff set class |
This definition is referenced by: casefun 6938 casedm 6939 caserel 6940 caseinj 6942 caseinl 6944 caseinr 6945 |
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