Definition df-inf 7044

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 7042	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4658	. . 3 class ◡𝑅
6	1, 2, 5	csup 7041	. 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  7070  infeq2  7073  infeq3  7074  infeq123d  7075  nfinf  7076  eqinfti  7079  infvalti  7081  infclti  7082  inflbti  7083  infglbti  7084  infsnti  7089  inf00  7090  infisoti  7091  infex2g  7093  dfinfre  8975  infrenegsupex  9659  infxrnegsupex  11406