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Theorem f1ssr 5305
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 5300 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
21adantr 274 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F  Fn  A )
3 simpr 109 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  ran  F  C_  C )
4 df-f 5097 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
52, 3, 4sylanbrc 413 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A
--> C )
6 df-f1 5098 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 273 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
87adantr 274 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  Fun  `' F
)
9 df-f1 5098 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
105, 8, 9sylanbrc 413 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 103    C_ wss 3041   `'ccnv 4508   ran crn 4510   Fun wfun 5087    Fn wfn 5088   -->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:  f1ff1  5306  difinfsn  6953
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