Definition df-inf 9483

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 9481	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5684	. . 3 class ◡𝑅
6	1, 2, 5	csup 9480	. 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  9516  infeq2  9519  infeq3  9520  infeq123d  9521  nfinf  9522  infexd  9523  eqinf  9524  infval  9526  infcl  9528  inflb  9529  infglb  9530  infglbb  9531  fiinfcl  9541  infltoreq  9542  inf00  9546  infempty  9547  infiso  9548  dfinfre  12249  infrenegsup  12251  tosglb  32965  rencldnfilem  42831