Definition df-inf 9132

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 9130	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5579	. . 3 class ◡𝑅
6	1, 2, 5	csup 9129	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1539	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9165  infeq2  9168  infeq3  9169  infeq123d  9170  nfinf  9171  infexd  9172  eqinf  9173  infval  9175  infcl  9177  inflb  9178  infglb  9179  infglbb  9180  fiinfcl  9190  infltoreq  9191  inf00  9195  infempty  9196  infiso  9197  dfinfre  11886  infrenegsup  11888  tosglb  31155  rencldnfilem  40558