Definition df-inf 7275

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 7273	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4747	. . 3 class ◡𝑅
6	1, 2, 5	csup 7272	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1398	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7301  infeq2  7304  infeq3  7305  infeq123d  7306  nfinf  7307  eqinfti  7310  infvalti  7312  infclti  7313  inflbti  7314  infglbti  7315  infsnti  7320  inf00  7321  infisoti  7322  infex2g  7324  dfinfre  9226  infrenegsupex  9922  infxrnegsupex  11941