Definition df-inf 9356

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 9354	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5630	. . 3 class ◡𝑅
6	1, 2, 5	csup 9353	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1542	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9390  infeq2  9393  infeq3  9394  infeq123d  9395  nfinf  9396  infexd  9397  eqinf  9398  infval  9400  infcl  9402  inflb  9403  infglb  9404  infglbb  9405  fiinfcl  9416  infltoreq  9417  inf00  9421  infempty  9422  infiso  9423  dfinfre  12137  infrenegsup  12139  tosglb  33035  rencldnfilem  43248