Definition df-case 7247

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 7245. 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 7246	. 2 class case(𝑅, 𝑆)
4		cinl 7208	. . . . 5 class inl
5	4	ccnv 4717	. . . 4 class ◡inl
6	1, 5	ccom 4722	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7209	. . . . 5 class inr
8	7	ccnv 4717	. . . 4 class ◡inr
9	2, 8	ccom 4722	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3195	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1395	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7248  casedm  7249  caserel  7250  caseinj  7252  caseinl  7254  caseinr  7255