Definition df-case 7145

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 7143. 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 7144	. 2 class case(𝑅, 𝑆)
4		cinl 7106	. . . . 5 class inl
5	4	ccnv 4659	. . . 4 class ◡inl
6	1, 5	ccom 4664	. . 3 class (𝑅 ∘ ◡inl)
7		cinr 7107	. . . . 5 class inr
8	7	ccnv 4659	. . . 4 class ◡inr
9	2, 8	ccom 4664	. . 3 class (𝑆 ∘ ◡inr)
10	6, 9	cun 3152	. 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  7146  casedm  7147  caserel  7148  caseinj  7150  caseinl  7152  caseinr  7153