Definition df-case 7431

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 7429. 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 7430	. 2 class case(𝑅, 𝑆)
4		cinl 7392	. . . . 5 class inl
5	4	ccnv 4775	. . . 4 class ◡inl
6	1, 5	ccom 4780	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7393	. . . . 5 class inr
8	7	ccnv 4775	. . . 4 class ◡inr
9	2, 8	ccom 4780	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3218	. 2 class ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
11	3, 10	wceq 1402	1 wff case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))

This definition is used by:  casefun  7432  casedm  7433  caserel  7434  caseinj  7436  caseinl  7438  caseinr  7439