Definition df-inf 9352

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 9350	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5622	. . 3 class ◡𝑅
6	1, 2, 5	csup 9349	. 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  9386  infeq2  9389  infeq3  9390  infeq123d  9391  nfinf  9392  infexd  9393  eqinf  9394  infval  9396  infcl  9398  inflb  9399  infglb  9400  infglbb  9401  fiinfcl  9412  infltoreq  9413  inf00  9417  infempty  9418  infiso  9419  dfinfre  12124  infrenegsup  12126  tosglb  32930  rencldnfilem  42793