Definition df-case 7377

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 7375. 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 7376	. 2 class case(𝑅, 𝑆)
4		cinl 7338	. . . . 5 class inl
5	4	ccnv 4750	. . . 4 class ◡inl
6	1, 5	ccom 4755	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7339	. . . . 5 class inr
8	7	ccnv 4750	. . . 4 class ◡inr
9	2, 8	ccom 4755	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3211	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1398	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7378  casedm  7379  caserel  7380  caseinj  7382  caseinl  7384  caseinr  7385