Definition df-inf 7094

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 7092	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4678	. . 3 class ◡𝑅
6	1, 2, 5	csup 7091	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1373	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7120  infeq2  7123  infeq3  7124  infeq123d  7125  nfinf  7126  eqinfti  7129  infvalti  7131  infclti  7132  inflbti  7133  infglbti  7134  infsnti  7139  inf00  7140  infisoti  7141  infex2g  7143  dfinfre  9036  infrenegsupex  9722  infxrnegsupex  11618