Definition df-case 7259

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 7257. 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 7258	. 2 class case(𝑅, 𝑆)
4		cinl 7220	. . . . 5 class inl
5	4	ccnv 4718	. . . 4 class ◡inl
6	1, 5	ccom 4723	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7221	. . . . 5 class inr
8	7	ccnv 4718	. . . 4 class ◡inr
9	2, 8	ccom 4723	. . 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  7260  casedm  7261  caserel  7262  caseinj  7264  caseinl  7266  caseinr  7267