Definition df-inf 8940

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 8938	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5523	. . 3 class ◡𝑅
6	1, 2, 5	csup 8937	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1538	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  8973  infeq2  8976  infeq3  8977  infeq123d  8978  nfinf  8979  infexd  8980  eqinf  8981  infval  8983  infcl  8985  inflb  8986  infglb  8987  infglbb  8988  fiinfcl  8998  infltoreq  8999  inf00  9003  infempty  9004  infiso  9005  dfinfre  11658  infrenegsup  11660  tosglb  30779  rencldnfilem  40134