MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1ssr Unicode version

Theorem f1ssr 5578
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 5573 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
21adantr 452 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F  Fn  A )
3 simpr 448 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  ran  F  C_  C )
4 df-f 5391 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
52, 3, 4sylanbrc 646 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A
--> C )
6 df-f1 5392 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 451 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
87adantr 452 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  Fun  `' F
)
9 df-f1 5392 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
105, 8, 9sylanbrc 646 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 359    C_ wss 3256   `'ccnv 4810   ran crn 4812   Fun wfun 5381    Fn wfn 5382   -->wf 5383   -1-1->wf1 5384
This theorem is referenced by:  domdifsn  7120  marypha1  7367  usgrares1  21283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361  df-f 5391  df-f1 5392
  Copyright terms: Public domain W3C validator