Definition df-case 7219

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 7217. 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 7218	. 2 class case(𝑅, 𝑆)
4		cinl 7180	. . . . 5 class inl
5	4	ccnv 4695	. . . 4 class ◡inl
6	1, 5	ccom 4700	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7181	. . . . 5 class inr
8	7	ccnv 4695	. . . 4 class ◡inr
9	2, 8	ccom 4700	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3175	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1375	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7220  casedm  7221  caserel  7222  caseinj  7224  caseinl  7226  caseinr  7227