Definition df-inf 9394

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 9392	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5637	. . 3 class ◡𝑅
6	1, 2, 5	csup 9391	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1540	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9428  infeq2  9431  infeq3  9432  infeq123d  9433  nfinf  9434  infexd  9435  eqinf  9436  infval  9438  infcl  9440  inflb  9441  infglb  9442  infglbb  9443  fiinfcl  9454  infltoreq  9455  inf00  9459  infempty  9460  infiso  9461  dfinfre  12164  infrenegsup  12166  tosglb  32901  rencldnfilem  42808