Definition df-case 6884

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 6882. 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 6883	. 2 class case(𝑅, 𝑆)
4		cinl 6845	. . . . 5 class inl
5	4	ccnv 4476	. . . 4 class ◡inl
6	1, 5	ccom 4481	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 6846	. . . . 5 class inr
8	7	ccnv 4476	. . . 4 class ◡inr
9	2, 8	ccom 4481	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3019	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1299	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  6885  casedm  6886  caserel  6887  caseinj  6889  caseinl  6891  caseinr  6892