Definition df-inf 7184

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 7182	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4724	. . 3 class ◡𝑅
6	1, 2, 5	csup 7181	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1397	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7210  infeq2  7213  infeq3  7214  infeq123d  7215  nfinf  7216  eqinfti  7219  infvalti  7221  infclti  7222  inflbti  7223  infglbti  7224  infsnti  7229  inf00  7230  infisoti  7231  infex2g  7233  dfinfre  9136  infrenegsupex  9828  infxrnegsupex  11825