Definition df-inf 7289

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 7287	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4753	. . 3 class ◡𝑅
6	1, 2, 5	csup 7286	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1398	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7315  infeq2  7318  infeq3  7319  infeq123d  7320  nfinf  7321  eqinfti  7324  infvalti  7326  infclti  7327  inflbti  7328  infglbti  7329  infsnti  7334  inf00  7335  infisoti  7336  infex2g  7338  dfinfre  9250  infrenegsupex  9947  infxrnegsupex  11976