Definition df-inf 7278

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 7276	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4750	. . 3 class ◡𝑅
6	1, 2, 5	csup 7275	. 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  7304  infeq2  7307  infeq3  7308  infeq123d  7309  nfinf  7310  eqinfti  7313  infvalti  7315  infclti  7316  inflbti  7317  infglbti  7318  infsnti  7323  inf00  7324  infisoti  7325  infex2g  7327  dfinfre  9235  infrenegsupex  9932  infxrnegsupex  11956