Definition df-case 7374

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 7372. 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 7373	. 2 class case(𝑅, 𝑆)
4		cinl 7335	. . . . 5 class inl
5	4	ccnv 4747	. . . 4 class ◡inl
6	1, 5	ccom 4752	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7336	. . . . 5 class inr
8	7	ccnv 4747	. . . 4 class ◡inr
9	2, 8	ccom 4752	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3208	. 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  7375  casedm  7376  caserel  7377  caseinj  7379  caseinl  7381  caseinr  7382