ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ssr Unicode version

Theorem f1ssr 5129
Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
f1ssr  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )

Proof of Theorem f1ssr
StepHypRef Expression
1 f1fn 5124 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
21adantr 270 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F  Fn  A )
3 simpr 108 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  ran  F  C_  C )
4 df-f 4936 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
52, 3, 4sylanbrc 408 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A
--> C )
6 df-f1 4937 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 269 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
87adantr 270 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  Fun  `' F
)
9 df-f1 4937 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
105, 8, 9sylanbrc 408 1  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    C_ wss 2974   `'ccnv 4370   ran crn 4372   Fun wfun 4926    Fn wfn 4927   -->wf 4928   -1-1->wf1 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-f 4936  df-f1 4937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator