ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ff1 Unicode version

Theorem f1ff1 5306
Description: If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)
Assertion
Ref Expression
f1ff1  |-  ( ( F : A -1-1-> B  /\  F : A --> C )  ->  F : A -1-1-> C )

Proof of Theorem f1ff1
StepHypRef Expression
1 frn 5251 . 2  |-  ( F : A --> C  ->  ran  F  C_  C )
2 f1ssr 5305 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )
31, 2sylan2 284 1  |-  ( ( F : A -1-1-> B  /\  F : A --> C )  ->  F : A -1-1-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    C_ wss 3041   ran crn 4510   -->wf 5089   -1-1->wf1 5090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-f 5097  df-f1 5098
This theorem is referenced by:  f1resf1  5308  inresflem  6913
  Copyright terms: Public domain W3C validator