Definition df-case 7425

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 7423. 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 7424	. 2 class case(𝑅, 𝑆)
4		cinl 7386	. . . . 5 class inl
5	4	ccnv 4773	. . . 4 class ◡inl
6	1, 5	ccom 4778	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7387	. . . . 5 class inr
8	7	ccnv 4773	. . . 4 class ◡inr
9	2, 8	ccom 4778	. . 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 used by:  casefun  7426  casedm  7427  caserel  7428  caseinj  7430  caseinl  7432  caseinr  7433