Definition df-inf 9211

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 9209	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5589	. . 3 class ◡𝑅
6	1, 2, 5	csup 9208	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1539	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  9244  infeq2  9247  infeq3  9248  infeq123d  9249  nfinf  9250  infexd  9251  eqinf  9252  infval  9254  infcl  9256  inflb  9257  infglb  9258  infglbb  9259  fiinfcl  9269  infltoreq  9270  inf00  9274  infempty  9275  infiso  9276  dfinfre  11965  infrenegsup  11967  tosglb  31262  rencldnfilem  40649