Definition df-inf 7325

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 7323	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4773	. . 3 class ◡𝑅
6	1, 2, 5	csup 7322	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1402	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is used by:  infeq1  7351  infeq2  7354  infeq3  7355  infeq123d  7356  nfinf  7357  eqinfti  7360  infvalti  7362  infclti  7363  inflbti  7364  infglbti  7365  infsnti  7370  inf00  7371  infisoti  7372  infex2g  7374  dfinfre  9287  infrenegsupex  9996  infxrnegsupex  12031