Definition df-case 7150

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 7148. 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 7149	. 2 class case(𝑅, 𝑆)
4		cinl 7111	. . . . 5 class inl
5	4	ccnv 4662	. . . 4 class ◡inl
6	1, 5	ccom 4667	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7112	. . . . 5 class inr
8	7	ccnv 4662	. . . 4 class ◡inr
9	2, 8	ccom 4667	. . 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  7151  casedm  7152  caserel  7153  caseinj  7155  caseinl  7157  caseinr  7158