Definition df-case 7061

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 7059. 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 7060	. 2 class case(𝑅, 𝑆)
4		cinl 7022	. . . . 5 class inl
5	4	ccnv 4610	. . . 4 class ◡inl
6	1, 5	ccom 4615	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7023	. . . . 5 class inr
8	7	ccnv 4610	. . . 4 class ◡inr
9	2, 8	ccom 4615	. . 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  7062  casedm  7063  caserel  7064  caseinj  7066  caseinl  7068  caseinr  7069