Definition df-case 7077

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 7075. 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 7076	. 2 class case(𝑅, 𝑆)
4		cinl 7038	. . . . 5 class inl
5	4	ccnv 4622	. . . 4 class ◡inl
6	1, 5	ccom 4627	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7039	. . . . 5 class inr
8	7	ccnv 4622	. . . 4 class ◡inr
9	2, 8	ccom 4627	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3127	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1353	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7078  casedm  7079  caserel  7080  caseinj  7082  caseinl  7084  caseinr  7085