Definition df-inf 9417

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 9415	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5658	. . 3 class ◡𝑅
6	1, 2, 5	csup 9414	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1570	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is used by:  infeq1  9451  infeq2  9454  infeq3  9455  infeq123d  9456  nfinf  9457  infexd  9458  eqinf  9459  infval  9461  infcl  9463  inflb  9464  infglb  9465  infglbb  9466  fiinfcl  9477  infltoreq  9478  inf00  9482  infempty  9483  infiso  9484  dfinfre  12224  infrenegsup  12226  tosglb  33423  rencldnfilem  43669