Definition df-inf 9438

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 9436	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5676	. . 3 class ◡𝑅
6	1, 2, 5	csup 9435	. 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  9471  infeq2  9474  infeq3  9475  infeq123d  9476  nfinf  9477  infexd  9478  eqinf  9479  infval  9481  infcl  9483  inflb  9484  infglb  9485  infglbb  9486  fiinfcl  9496  infltoreq  9497  inf00  9501  infempty  9502  infiso  9503  dfinfre  12195  infrenegsup  12197  tosglb  32176  rencldnfilem  41606