Definition df-case 6969

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 6967. 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 6968	. 2 class case(𝑅, 𝑆)
4		cinl 6930	. . . . 5 class inl
5	4	ccnv 4538	. . . 4 class ◡inl
6	1, 5	ccom 4543	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 6931	. . . . 5 class inr
8	7	ccnv 4538	. . . 4 class ◡inr
9	2, 8	ccom 4543	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3069	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1331	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is referenced by:  casefun  6970  casedm  6971  caserel  6972  caseinj  6974  caseinl  6976  caseinr  6977