Definition df-case 7117

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 7115. 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 7116	. 2 class case(𝑅, 𝑆)
4		cinl 7078	. . . . 5 class inl
5	4	ccnv 4646	. . . 4 class ◡inl
6	1, 5	ccom 4651	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7079	. . . . 5 class inr
8	7	ccnv 4646	. . . 4 class ◡inr
9	2, 8	ccom 4651	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3142	. 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  7118  casedm  7119  caserel  7120  caseinj  7122  caseinl  7124  caseinr  7125