Definition df-case 7040

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 7038. 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 7039	. 2 class case(𝑅, 𝑆)
4		cinl 7001	. . . . 5 class inl
5	4	ccnv 4597	. . . 4 class ◡inl
6	1, 5	ccom 4602	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7002	. . . . 5 class inr
8	7	ccnv 4597	. . . 4 class ◡inr
9	2, 8	ccom 4602	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3109	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1342	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  7041  casedm  7042  caserel  7043  caseinj  7045  caseinl  7047  caseinr  7048