Definition df-inf 9402

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 9400	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5660	. . 3 class ◡𝑅
6	1, 2, 5	csup 9399	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1568	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9436  infeq2  9439  infeq3  9440  infeq123d  9441  nfinf  9442  infexd  9443  eqinf  9444  infval  9446  infcl  9448  inflb  9449  infglb  9450  infglbb  9451  fiinfcl  9462  infltoreq  9463  inf00  9467  infempty  9468  infiso  9469  dfinfre  12195  infrenegsup  12197  tosglb  33261  rencldnfilem  43517