Definition df-inf 7227

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 7225	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4730	. . 3 class ◡𝑅
6	1, 2, 5	csup 7224	. 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  7253  infeq2  7256  infeq3  7257  infeq123d  7258  nfinf  7259  eqinfti  7262  infvalti  7264  infclti  7265  inflbti  7266  infglbti  7267  infsnti  7272  inf00  7273  infisoti  7274  infex2g  7276  dfinfre  9179  infrenegsupex  9873  infxrnegsupex  11886