Definition df-case 7388

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 7386. 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 7387	. 2 class case(𝑅, 𝑆)
4		cinl 7349	. . . . 5 class inl
5	4	ccnv 4753	. . . 4 class ◡inl
6	1, 5	ccom 4758	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7350	. . . . 5 class inr
8	7	ccnv 4753	. . . 4 class ◡inr
9	2, 8	ccom 4758	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3212	. 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  7389  casedm  7390  caserel  7391  caseinj  7393  caseinl  7395  caseinr  7396