Definition df-case 7059

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 7057. 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 7058	. 2 class case(𝑅, 𝑆)
4		cinl 7020	. . . . 5 class inl
5	4	ccnv 4608	. . . 4 class ◡inl
6	1, 5	ccom 4613	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7021	. . . . 5 class inr
8	7	ccnv 4608	. . . 4 class ◡inr
9	2, 8	ccom 4613	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3119	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1348	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7060  casedm  7061  caserel  7062  caseinj  7064  caseinl  7066  caseinr  7067