Definition df-inf 9480

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 9478	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5687	. . 3 class ◡𝑅
6	1, 2, 5	csup 9477	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1536	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9513  infeq2  9516  infeq3  9517  infeq123d  9518  nfinf  9519  infexd  9520  eqinf  9521  infval  9523  infcl  9525  inflb  9526  infglb  9527  infglbb  9528  fiinfcl  9538  infltoreq  9539  inf00  9543  infempty  9544  infiso  9545  dfinfre  12246  infrenegsup  12248  tosglb  32949  rencldnfilem  42807