Definition df-case 7143

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 7141. 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 7142	. 2 class case(𝑅, 𝑆)
4		cinl 7104	. . . . 5 class inl
5	4	ccnv 4658	. . . 4 class ◡inl
6	1, 5	ccom 4663	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7105	. . . . 5 class inr
8	7	ccnv 4658	. . . 4 class ◡inr
9	2, 8	ccom 4663	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3151	. 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  7144  casedm  7145  caserel  7146  caseinj  7148  caseinl  7150  caseinr  7151