Definition df-inf 9353

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 9351	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5624	. . 3 class ◡𝑅
6	1, 2, 5	csup 9350	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1547	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9387  infeq2  9390  infeq3  9391  infeq123d  9392  nfinf  9393  infexd  9394  eqinf  9395  infval  9397  infcl  9399  inflb  9400  infglb  9401  infglbb  9402  fiinfcl  9413  infltoreq  9414  inf00  9418  infempty  9419  infiso  9420  dfinfre  12135  infrenegsup  12137  tosglb  33061  rencldnfilem  43266