Definition df-inf 7319

Description: Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)

Assertion

Ref	Expression
df-inf	⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

Detailed syntax breakdown of Definition df-inf

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		cR	. . 3 class 𝑅
4	1, 2, 3	cinf 7317	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4771	. . 3 class ◡𝑅
6	1, 2, 5	csup 7316	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1402	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7345  infeq2  7348  infeq3  7349  infeq123d  7350  nfinf  7351  eqinfti  7354  infvalti  7356  infclti  7357  inflbti  7358  infglbti  7359  infsnti  7364  inf00  7365  infisoti  7366  infex2g  7368  dfinfre  9280  infrenegsupex  9977  infxrnegsupex  12012