Definition df-inf 7120

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 7118	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4695	. . 3 class ◡𝑅
6	1, 2, 5	csup 7117	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1375	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7146  infeq2  7149  infeq3  7150  infeq123d  7151  nfinf  7152  eqinfti  7155  infvalti  7157  infclti  7158  inflbti  7159  infglbti  7160  infsnti  7165  inf00  7166  infisoti  7167  infex2g  7169  dfinfre  9071  infrenegsupex  9757  infxrnegsupex  11740