Definition df-case 7277

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 7275. 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 7276	. 2 class case(𝑅, 𝑆)
4		cinl 7238	. . . . 5 class inl
5	4	ccnv 4722	. . . 4 class ◡inl
6	1, 5	ccom 4727	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7239	. . . . 5 class inr
8	7	ccnv 4722	. . . 4 class ◡inr
9	2, 8	ccom 4727	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3196	. 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  7278  casedm  7279  caserel  7280  caseinj  7282  caseinl  7284  caseinr  7285