Definition df-inf 9512

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 9510	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5699	. . 3 class ◡𝑅
6	1, 2, 5	csup 9509	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1537	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9545  infeq2  9548  infeq3  9549  infeq123d  9550  nfinf  9551  infexd  9552  eqinf  9553  infval  9555  infcl  9557  inflb  9558  infglb  9559  infglbb  9560  fiinfcl  9570  infltoreq  9571  inf00  9575  infempty  9576  infiso  9577  dfinfre  12276  infrenegsup  12278  tosglb  32948  rencldnfilem  42776