Definition df-inf 7326

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 7324	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4773	. . 3 class ◡𝑅
6	1, 2, 5	csup 7323	. 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  7352  infeq2  7355  infeq3  7356  infeq123d  7357  nfinf  7358  eqinfti  7361  infvalti  7363  infclti  7364  inflbti  7365  infglbti  7366  infsnti  7371  inf00  7372  infisoti  7373  infex2g  7375  dfinfre  9289  infrenegsupex  10004  infxrnegsupex  12047