Definition df-inf 7113

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 7111	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4692	. . 3 class ◡𝑅
6	1, 2, 5	csup 7110	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1373	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7139  infeq2  7142  infeq3  7143  infeq123d  7144  nfinf  7145  eqinfti  7148  infvalti  7150  infclti  7151  inflbti  7152  infglbti  7153  infsnti  7158  inf00  7159  infisoti  7160  infex2g  7162  dfinfre  9064  infrenegsupex  9750  infxrnegsupex  11689