Definition df-inf 9358

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 9356	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5631	. . 3 class ◡𝑅
6	1, 2, 5	csup 9355	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1542	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9392  infeq2  9395  infeq3  9396  infeq123d  9397  nfinf  9398  infexd  9399  eqinf  9400  infval  9402  infcl  9404  inflb  9405  infglb  9406  infglbb  9407  fiinfcl  9418  infltoreq  9419  inf00  9423  infempty  9424  infiso  9425  dfinfre  12135  infrenegsup  12137  tosglb  33067  rencldnfilem  43166