Definition df-inf 9453

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 9451	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5653	. . 3 class ◡𝑅
6	1, 2, 5	csup 9450	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1540	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9487  infeq2  9490  infeq3  9491  infeq123d  9492  nfinf  9493  infexd  9494  eqinf  9495  infval  9497  infcl  9499  inflb  9500  infglb  9501  infglbb  9502  fiinfcl  9513  infltoreq  9514  inf00  9518  infempty  9519  infiso  9520  dfinfre  12221  infrenegsup  12223  tosglb  32901  rencldnfilem  42790