Definition df-inf 9346

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 9344	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5623	. . 3 class ◡𝑅
6	1, 2, 5	csup 9343	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1541	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9380  infeq2  9383  infeq3  9384  infeq123d  9385  nfinf  9386  infexd  9387  eqinf  9388  infval  9390  infcl  9392  inflb  9393  infglb  9394  infglbb  9395  fiinfcl  9406  infltoreq  9407  inf00  9411  infempty  9412  infiso  9413  dfinfre  12123  infrenegsup  12125  tosglb  33057  rencldnfilem  43058