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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 7115. 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 7116 | . 2 class case(𝑅, 𝑆) |
4 | cinl 7078 | . . . . 5 class inl | |
5 | 4 | ccnv 4646 | . . . 4 class ◡inl |
6 | 1, 5 | ccom 4651 | . . 3 class (𝑅 ∘ ◡inl) |
7 | cinr 7079 | . . . . 5 class inr | |
8 | 7 | ccnv 4646 | . . . 4 class ◡inr |
9 | 2, 8 | ccom 4651 | . . 3 class (𝑆 ∘ ◡inr) |
10 | 6, 9 | cun 3142 | . 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
11 | 3, 10 | wceq 1364 | 1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) |
Colors of variables: wff set class |
This definition is referenced by: casefun 7118 casedm 7119 caserel 7120 caseinj 7122 caseinl 7124 caseinr 7125 |
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