Definition df-case 7430

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 7428. 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 7429	. 2 class case(𝑅, 𝑆)
4		cinl 7391	. . . . 5 class inl
5	4	ccnv 4775	. . . 4 class ◡inl
6	1, 5	ccom 4780	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7392	. . . . 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  7431  casedm  7432  caserel  7433  caseinj  7435  caseinl  7437  caseinr  7438