Definition df-inf 9388

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 9386	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5637	. . 3 class ◡𝑅
6	1, 2, 5	csup 9385	. 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  9421  infeq2  9424  infeq3  9425  infeq123d  9426  nfinf  9427  infexd  9428  eqinf  9429  infval  9431  infcl  9433  inflb  9434  infglb  9435  infglbb  9436  fiinfcl  9446  infltoreq  9447  inf00  9451  infempty  9452  infiso  9453  dfinfre  12145  infrenegsup  12147  tosglb  31905  rencldnfilem  41201