Definition df-inf 9438

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 9436	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5649	. . 3 class ◡𝑅
6	1, 2, 5	csup 9435	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1569	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is used by:  infeq1  9472  infeq2  9475  infeq3  9476  infeq123d  9477  nfinf  9478  infexd  9479  eqinf  9480  infval  9482  infcl  9484  inflb  9485  infglb  9486  infglbb  9487  fiinfcl  9498  infltoreq  9499  inf00  9503  infempty  9504  infiso  9505  dfinfre  12301  infrenegsup  12303  tosglb  33582  rencldnfilem  43920