Definition df-inf 8907

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 8905	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5554	. . 3 class ◡𝑅
6	1, 2, 5	csup 8904	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1537	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  8940  infeq2  8943  infeq3  8944  infeq123d  8945  nfinf  8946  infexd  8947  eqinf  8948  infval  8950  infcl  8952  inflb  8953  infglb  8954  infglbb  8955  fiinfcl  8965  infltoreq  8966  inf00  8970  infempty  8971  infiso  8972  dfinfre  11622  infrenegsup  11624  tosglb  30657  rencldnfilem  39466