Definition df-inf 8232

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 8230	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5037	. . 3 class ◡𝑅
6	1, 2, 5	csup 8229	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1475	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  8265  infeq2  8268  infeq3  8269  infeq123d  8270  nfinf  8271  infexd  8272  eqinf  8273  infval  8275  infcl  8277  inflb  8278  infglb  8279  infglbb  8280  fiinfcl  8290  infltoreq  8291  inf00  8294  infempty  8295  infiso  8296  lbinf  10855  dfinfre  10881  infrenegsup  10883  tosglb  29001  rencldnfilem  36402