Definition df-inf 9347

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 9345	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5621	. . 3 class ◡𝑅
6	1, 2, 5	csup 9344	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1542	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9381  infeq2  9384  infeq3  9385  infeq123d  9386  nfinf  9387  infexd  9388  eqinf  9389  infval  9391  infcl  9393  inflb  9394  infglb  9395  infglbb  9396  fiinfcl  9407  infltoreq  9408  inf00  9412  infempty  9413  infiso  9414  dfinfre  12126  infrenegsup  12128  tosglb  33055  rencldnfilem  43263