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Theorem f1ssr 5802
Description: A function that is one-to-one is also one-to-one on some superset of its range. (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 5797 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
21adantr 466 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F  Fn  A )
3 simpr 462 . . 3  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  ran  F  C_  C )
4 df-f 5605 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
52, 3, 4sylanbrc 668 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A
--> C )
6 df-f1 5606 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
76simprbi 465 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
87adantr 466 . 2  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  Fun  `' F
)
9 df-f1 5606 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
105, 8, 9sylanbrc 668 1  |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C
)  ->  F : A -1-1-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    C_ wss 3442   `'ccnv 4853   ran crn 4855   Fun wfun 5595    Fn wfn 5596   -->wf 5597   -1-1->wf1 5598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372  df-f 5605  df-f1 5606
This theorem is referenced by:  domdifsn  7661  marypha1  7954  m2cpmf1  19698  usgrares1  24983  usgresvm1  38513  usgresvm1ALT  38517
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