Definition df-inf 7051

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 7049	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4662	. . 3 class ◡𝑅
6	1, 2, 5	csup 7048	. 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  7077  infeq2  7080  infeq3  7081  infeq123d  7082  nfinf  7083  eqinfti  7086  infvalti  7088  infclti  7089  inflbti  7090  infglbti  7091  infsnti  7096  inf00  7097  infisoti  7098  infex2g  7100  dfinfre  8983  infrenegsupex  9668  infxrnegsupex  11428