Definition df-inf 7189

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 7187	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4726	. . 3 class ◡𝑅
6	1, 2, 5	csup 7186	. 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  7215  infeq2  7218  infeq3  7219  infeq123d  7220  nfinf  7221  eqinfti  7224  infvalti  7226  infclti  7227  inflbti  7228  infglbti  7229  infsnti  7234  inf00  7235  infisoti  7236  infex2g  7238  dfinfre  9141  infrenegsupex  9833  infxrnegsupex  11846