Definition df-case 7288

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 7286. 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 7287	. 2 class case(𝑅, 𝑆)
4		cinl 7249	. . . . 5 class inl
5	4	ccnv 4726	. . . 4 class ◡inl
6	1, 5	ccom 4731	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7250	. . . . 5 class inr
8	7	ccnv 4726	. . . 4 class ◡inr
9	2, 8	ccom 4731	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3197	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1397	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7289  casedm  7290  caserel  7291  caseinj  7293  caseinl  7295  caseinr  7296