Definition df-inf 9419

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 9417	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5650	. . 3 class ◡𝑅
6	1, 2, 5	csup 9416	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1570	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is used by:  infeq1  9453  infeq2  9456  infeq3  9457  infeq123d  9458  nfinf  9459  infexd  9460  eqinf  9461  infval  9463  infcl  9465  inflb  9466  infglb  9467  infglbb  9468  fiinfcl  9479  infltoreq  9480  inf00  9484  infempty  9485  infiso  9486  dfinfre  12276  infrenegsup  12278  tosglb  33515  rencldnfilem  43777