Definition df-case 7267

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 7265. 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.)

Assertion

Ref	Expression
df-case	⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

Detailed syntax breakdown of Definition df-case

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2		cS	. . 3 class 𝑆
3	1, 2	cdjucase 7266	. 2 class case(𝑅, 𝑆)
4		cinl 7228	. . . . 5 class inl
5	4	ccnv 4719	. . . 4 class ◡inl
6	1, 5	ccom 4724	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7229	. . . . 5 class inr
8	7	ccnv 4719	. . . 4 class ◡inr
9	2, 8	ccom 4724	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3195	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1395	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7268  casedm  7269  caserel  7270  caseinj  7272  caseinl  7274  caseinr  7275