Definition df-inf 8618

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 8616	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 5341	. . 3 class ◡𝑅
6	1, 2, 5	csup 8615	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1656	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  8651  infeq2  8654  infeq3  8655  infeq123d  8656  nfinf  8657  infexd  8658  eqinf  8659  infval  8661  infcl  8663  inflb  8664  infglb  8665  infglbb  8666  fiinfcl  8676  infltoreq  8677  inf00  8680  infempty  8681  infiso  8682  dfinfre  11334  infrenegsup  11336  tosglb  30204  rencldnfilem  38221