Definition df-case 7159

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 7157. 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 7158	. 2 class case(𝑅, 𝑆)
4		cinl 7120	. . . . 5 class inl
5	4	ccnv 4663	. . . 4 class ◡inl
6	1, 5	ccom 4668	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7121	. . . . 5 class inr
8	7	ccnv 4663	. . . 4 class ◡inr
9	2, 8	ccom 4668	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3155	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1364	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7160  casedm  7161  caserel  7162  caseinj  7164  caseinl  7166  caseinr  7167