Definition df-inf 9435

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 9433	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5675	. . 3 class ◡𝑅
6	1, 2, 5	csup 9432	. 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  9468  infeq2  9471  infeq3  9472  infeq123d  9473  nfinf  9474  infexd  9475  eqinf  9476  infval  9478  infcl  9480  inflb  9481  infglb  9482  infglbb  9483  fiinfcl  9493  infltoreq  9494  inf00  9498  infempty  9499  infiso  9500  dfinfre  12192  infrenegsup  12194  tosglb  32133  rencldnfilem  41544