Definition df-case 7085

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 7083. 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 7084	. 2 class case(𝑅, 𝑆)
4		cinl 7046	. . . . 5 class inl
5	4	ccnv 4627	. . . 4 class ◡inl
6	1, 5	ccom 4632	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7047	. . . . 5 class inr
8	7	ccnv 4627	. . . 4 class ◡inr
9	2, 8	ccom 4632	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3129	. 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  7086  casedm  7087  caserel  7088  caseinj  7090  caseinl  7092  caseinr  7093