Definition df-case 7418

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 7416. 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 7417	. 2 class case(𝑅, 𝑆)
4		cinl 7379	. . . . 5 class inl
5	4	ccnv 4771	. . . 4 class ◡inl
6	1, 5	ccom 4776	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7380	. . . . 5 class inr
8	7	ccnv 4771	. . . 4 class ◡inr
9	2, 8	ccom 4776	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3218	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1402	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7419  casedm  7420  caserel  7421  caseinj  7423  caseinl  7425  caseinr  7426