Definition df-inf 9350

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 9348	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5624	. . 3 class ◡𝑅
6	1, 2, 5	csup 9347	. 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  9384  infeq2  9387  infeq3  9388  infeq123d  9389  nfinf  9390  infexd  9391  eqinf  9392  infval  9394  infcl  9396  inflb  9397  infglb  9398  infglbb  9399  fiinfcl  9410  infltoreq  9411  inf00  9415  infempty  9416  infiso  9417  dfinfre  12127  infrenegsup  12129  tosglb  33038  rencldnfilem  43098