Definition df-inf 9384

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 9382	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5644	. . 3 class ◡𝑅
6	1, 2, 5	csup 9381	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1559	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9418  infeq2  9421  infeq3  9422  infeq123d  9423  nfinf  9424  infexd  9425  eqinf  9426  infval  9428  infcl  9430  inflb  9431  infglb  9432  infglbb  9433  fiinfcl  9444  infltoreq  9445  inf00  9449  infempty  9450  infiso  9451  dfinfre  12168  infrenegsup  12170  tosglb  33112  rencldnfilem  43350