Definition df-inf 7160

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 7158	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4718	. . 3 class ◡𝑅
6	1, 2, 5	csup 7157	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1395	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7186  infeq2  7189  infeq3  7190  infeq123d  7191  nfinf  7192  eqinfti  7195  infvalti  7197  infclti  7198  inflbti  7199  infglbti  7200  infsnti  7205  inf00  7206  infisoti  7207  infex2g  7209  dfinfre  9111  infrenegsupex  9797  infxrnegsupex  11782