Definition df-case 7193

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 7191. 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 7192	. 2 class case(𝑅, 𝑆)
4		cinl 7154	. . . . 5 class inl
5	4	ccnv 4678	. . . 4 class ◡inl
6	1, 5	ccom 4683	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7155	. . . . 5 class inr
8	7	ccnv 4678	. . . 4 class ◡inr
9	2, 8	ccom 4683	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3165	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1373	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7194  casedm  7195  caserel  7196  caseinj  7198  caseinl  7200  caseinr  7201