Definition df-inf 7178

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 7176	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4722	. . 3 class ◡𝑅
6	1, 2, 5	csup 7175	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1395	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7204  infeq2  7207  infeq3  7208  infeq123d  7209  nfinf  7210  eqinfti  7213  infvalti  7215  infclti  7216  inflbti  7217  infglbti  7218  infsnti  7223  inf00  7224  infisoti  7225  infex2g  7227  dfinfre  9129  infrenegsupex  9821  infxrnegsupex  11817