Definition df-inf 6986

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 6984	. 2 class inf(𝐴, 𝐵, 𝑅)
5	3	ccnv 4627	. . 3 class ◡𝑅
6	1, 2, 5	csup 6983	. 2 class sup(𝐴, 𝐵, ◡𝑅)
7	4, 6	wceq 1353	1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)

This definition is referenced by:  infeq1  7012  infeq2  7015  infeq3  7016  infeq123d  7017  nfinf  7018  eqinfti  7021  infvalti  7023  infclti  7024  inflbti  7025  infglbti  7026  infsnti  7031  inf00  7032  infisoti  7033  infex2g  7035  dfinfre  8915  infrenegsupex  9596  infxrnegsupex  11273