Definition df-inf 9322

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 9320	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5613	. . 3 class ◡𝑅
6	1, 2, 5	csup 9319	. 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  9356  infeq2  9359  infeq3  9360  infeq123d  9361  nfinf  9362  infexd  9363  eqinf  9364  infval  9366  infcl  9368  inflb  9369  infglb  9370  infglbb  9371  fiinfcl  9382  infltoreq  9383  inf00  9387  infempty  9388  infiso  9389  dfinfre  12095  infrenegsup  12097  tosglb  32946  rencldnfilem  42832