Definition df-inf 7046

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 7044	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4659	. . 3 class ◡𝑅
6	1, 2, 5	csup 7043	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1364	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7072  infeq2  7075  infeq3  7076  infeq123d  7077  nfinf  7078  eqinfti  7081  infvalti  7083  infclti  7084  inflbti  7085  infglbti  7086  infsnti  7091  inf00  7092  infisoti  7093  infex2g  7095  dfinfre  8977  infrenegsupex  9662  infxrnegsupex  11409