Definition df-inf 9413

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 9411	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5665	. . 3 class ◡𝑅
6	1, 2, 5	csup 9410	. 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  9447  infeq2  9450  infeq3  9451  infeq123d  9452  nfinf  9453  infexd  9454  eqinf  9455  infval  9457  infcl  9459  inflb  9460  infglb  9461  infglbb  9462  fiinfcl  9473  infltoreq  9474  inf00  9478  infempty  9479  infiso  9480  dfinfre  12214  infrenegsup  12216  tosglb  33319  rencldnfilem  43580