Definition df-inf 9337

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 9335	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5620	. . 3 class ◡𝑅
6	1, 2, 5	csup 9334	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1541	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9371  infeq2  9374  infeq3  9375  infeq123d  9376  nfinf  9377  infexd  9378  eqinf  9379  infval  9381  infcl  9383  inflb  9384  infglb  9385  infglbb  9386  fiinfcl  9397  infltoreq  9398  inf00  9402  infempty  9403  infiso  9404  dfinfre  12113  infrenegsup  12115  tosglb  32967  rencldnfilem  42927