Definition df-inf 7168

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 7166	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4719	. . 3 class ◡𝑅
6	1, 2, 5	csup 7165	. 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  7194  infeq2  7197  infeq3  7198  infeq123d  7199  nfinf  7200  eqinfti  7203  infvalti  7205  infclti  7206  inflbti  7207  infglbti  7208  infsnti  7213  inf00  7214  infisoti  7215  infex2g  7217  dfinfre  9119  infrenegsupex  9806  infxrnegsupex  11795