Definition df-inf 6962

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 6960	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4610	. . 3 class ◡𝑅
6	1, 2, 5	csup 6959	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1348	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  6988  infeq2  6991  infeq3  6992  infeq123d  6993  nfinf  6994  eqinfti  6997  infvalti  6999  infclti  7000  inflbti  7001  infglbti  7002  infsnti  7007  inf00  7008  infisoti  7009  infex2g  7011  dfinfre  8872  infrenegsupex  9553  infxrnegsupex  11226