Metamath Proof Explorer


Theorem isopo

Description: An isomorphism preserves the property of being a partial order. (Contributed by Stefan O'Rear, 16-Nov-2014)

Ref Expression
Assertion isopo ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Po 𝐴𝑆 Po 𝐵 ) )

Proof

Step Hyp Ref Expression
1 isocnv ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) )
2 isopolem ( 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) → ( 𝑅 Po 𝐴𝑆 Po 𝐵 ) )
3 1 2 syl ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Po 𝐴𝑆 Po 𝐵 ) )
4 isopolem ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑆 Po 𝐵𝑅 Po 𝐴 ) )
5 3 4 impbid ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Po 𝐴𝑆 Po 𝐵 ) )