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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 6882. 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 6883 | . 2 class case(𝑅, 𝑆) |
4 | cinl 6845 | . . . . 5 class inl | |
5 | 4 | ccnv 4476 | . . . 4 class ◡inl |
6 | 1, 5 | ccom 4481 | . . 3 class (𝑅 ∘ ◡inl) |
7 | cinr 6846 | . . . . 5 class inr | |
8 | 7 | ccnv 4476 | . . . 4 class ◡inr |
9 | 2, 8 | ccom 4481 | . . 3 class (𝑆 ∘ ◡inr) |
10 | 6, 9 | cun 3019 | . 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
11 | 3, 10 | wceq 1299 | 1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
Colors of variables: wff set class |
This definition is referenced by: casefun 6885 casedm 6886 caserel 6887 caseinj 6889 caseinl 6891 caseinr 6892 |
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