Definition df-inf 9462

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 9460	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5654	. . 3 class ◡𝑅
6	1, 2, 5	csup 9459	. 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  9496  infeq2  9499  infeq3  9500  infeq123d  9501  nfinf  9502  infexd  9503  eqinf  9504  infval  9506  infcl  9508  inflb  9509  infglb  9510  infglbb  9511  fiinfcl  9522  infltoreq  9523  inf00  9527  infempty  9528  infiso  9529  dfinfre  12325  infrenegsup  12327  tosglb  33606  rencldnfilem  44020