Definition df-inf 7332

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 7330	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4775	. . 3 class ◡𝑅
6	1, 2, 5	csup 7329	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1402	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is used by:  infeq1  7358  infeq2  7361  infeq3  7362  infeq123d  7363  nfinf  7364  eqinfti  7367  infvalti  7369  infclti  7370  inflbti  7371  infglbti  7372  infsnti  7377  inf00  7378  infisoti  7379  infex2g  7381  dfinfre  9295  infrenegsupex  10011  infxrnegsupex  12055